17: Microbial Mechanisms of Pathogenicity
Jane woke up one spring morning feeling not quite herself. Her throat felt a bit dry and she was sniffling. She wondered why she felt so lousy. Was it because of a change in the weather? The pollen count? Was she coming down with something? Did she catch a bug from her coworker who sneezed on her in the elevator yesterday?
The signs and symptoms we associate with illness can have many different causes. Sometimes they are the direct result of a pathogenic infection, but in other cases they result from a response by our immune system to a pathogen or another perceived threat. For example, in response to certain pathogens, the immune system may release pyrogens, chemicals that cause the body temperature to rise, resulting in a fever. This response creates a less-than-favorable environment for the pathogen, but it also makes us feel sick.
Medical professionals rely heavily on analysis of signs and symptoms to determine the cause of an ailment and prescribe treatment. In some cases, signs and symptoms alone are enough to correctly identify the causative agent of a disease, but since few diseases produce truly unique symptoms, it is often necessary to confirm the identity of the infectious agent by other direct and indirect diagnostic methods.
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- 17.1: Characteristics of Infectious Diseases
- In an infection, a microorganism enters a host and begins to multiply. Some infections cause disease, which is any deviation from the normal function or structure of the host. Signs of a disease are objective and are measured. Symptoms of a disease are subjective and are reported by the patient. Diseases can either be noninfectious (due to genetics and environment) or infectious (due to pathogens).
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- 17.2: How Pathogens Cause Disease
- Koch’s postulates are used to determine whether a particular microorganism is a pathogen. Molecular Koch’s postulates are used to determine what genes contribute to a pathogen’s ability to cause disease. Virulence, the degree to which a pathogen can cause disease, can be quantified by calculating either the ID50 or LD50 of a pathogen on a given population. Primary pathogens are capable of causing pathological changes associated with disease in a healthy individual.
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- 17.3: Virulence Factors
- Virulence factors contribute to a pathogen’s ability to cause disease. Exoenzymes and toxins allow pathogens to invade host tissue and cause tissue damage. Exoenzymes are classified according to the macromolecule they target and exotoxins are classified based on their mechanism of action. Bacterial toxins include endotoxin and exotoxins. Endotoxin is the lipid A component of the LPS of the gram-negative cell envelope. Exotoxins are proteins secreted mainly by gram-positive bacteria.
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- 17.4: Aseptic Techniques
- Fungal and parasitic pathogens use pathogenic mechanisms and virulence factors that are similar to those of bacterial pathogens. Fungi initiate infections through the interaction of adhesins with receptors on host cells. Some fungi produce toxins and exoenzymes involved in disease production and capsules that provide protection of phagocytosis. Protozoa adhere to target cells through complex mechanisms and can cause cellular damage through release of cytopathic substances.
Thumbnail: Ulcer-causing bacterium ( H. Pylori ) crossing mucus layer of stomach. (Public Domain/modified from original; National Science Foundation via Wikimedia Commons ).


18.E: Electric Charge and Electric Field (Exercises)
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Conceptual Questions
18.1: Static Electricity and Charge: Conservation of Charge
1 . There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
2. Why do most objects tend to contain nearly equal numbers of positive and negative charges?
18.2: Conductors and Insulators
3. An eccentric inventor attempts to levitate by first placing a large negative charge on himself and then putting a large positive charge on the ceiling of his workshop.  Explain.
4. If you have charged an electroscope by contact with a positively charged object, describe how you could use it to determine the charge of other objects. Specifically, what would the leaves of the electroscope do if other charged objects were brought near its knob?
5. When a glass rod is rubbed with silk, it becomes positive and the silk becomes negative—yet both attract dust. Does the dust have a third type of charge that is attracted to both positive and negative? Explain.
6. Why does a car always attract dust right after it is polished? (Note that car wax and car tires are insulators.)
7. Describe how a positively charged object can be used to give another object a negative charge. What is the name of this process?
8. What is grounding? What effect does it have on a charged conductor? On a charged insulator?
18.3: Coulomb's Law
9. Figure shows the charge distribution in a water molecule, which is called a polar molecule because it has an inherent separation of charge. Given water’s polar character, explain what effect humidity has on removing excess charge from objects.
Schematic representation of the outer electron cloud of a neutral water molecule. The electrons spend more time near the oxygen than the hydrogens, giving a permanent charge separation as shown. Water is thus a polar molecule. It is more easily affected by electrostatic forces than molecules with uniform charge distributions.
10. Using Figure, explain, in terms of Coulomb’s law, why a polar molecule (such as in Figure) is attracted by both positive and negative charges.
11. Given the polar character of water molecules, explain how ions in the air form nucleation centers for rain droplets.
18.4: Electric Field: Concept of a Field Revisited
12. Why must the test charge \(q\) in the definition of the electric field be vanishing small?
13. Are the direction and magnitude of the Coulomb force unique at a given point in space? What about the electric field?
18.5: Electric Field Lines: Multiple Charges
14. Compare and contrast the Coulomb force field and the electric field. To do this, make a list of five properties for the Coulomb force field analogous to the five properties listed for electric field lines. Compare each item in your list of Coulomb force field properties with those of the electric field—are they the same or different? (For example, electric field lines cannot cross. Is the same true for Coulomb field lines?)
15. Figure shows an electric field extending over three regions, labeled I, II, and III. Answer the following questions.
(a) Are there any isolated charges? If so, in what region and what are their signs?
(b) Where is the field strongest?
(c) Where is it weakest?
(d) Where is the field the most uniform?
18.6: Electric Forces in Biology
16. A cell membrane is a thin layer enveloping a cell. The thickness of the membrane is much less than the size of the cell. In a static situation the membrane has a charge distribution of \(-2.5\times 10^{-6} C/m^{2}\) on its inner surface and \(+2.5\times 10^{-6}C/m^{2}\) on its outer surface. Draw a diagram of the cell and the surrounding cell membrane. Include on this diagram the charge distribution and the corresponding electric field. Is there any electric field inside the cell? Is there any electric field outside the cell?
18.7: Conductors and Electric Fields in Static Equilibrium
17. Is the object in Figure 18.8.9 a conductor or an insulator? Justify your answer.
18. If the electric field lines in the figure above were perpendicular to the object, would it necessarily be a conductor? Explain.
19. The discussion of the electric field between two parallel conducting plates, in this module states that edge effects are less important if the plates are close together. What does close mean? That is, is the actual plate separation crucial, or is the ratio of plate separation to plate area crucial?
20. Would the self-created electric field at the end of a pointed conductor, such as a lightning rod, remove positive or negative charge from the conductor? Would the same sign charge be removed from a neutral pointed conductor by the application of a similar externally created electric field? (The answers to both questions have implications for charge transfer utilizing points.)
21. Why is a golfer with a metal club over her shoulder vulnerable to lightning in an open fairway? Would she be any safer under a tree?
22. Can the belt of a Van de Graaff accelerator be a conductor? Explain.
23. Are you relatively safe from lightning inside an automobile? Give two reasons.
24. Discuss pros and cons of a lightning rod being grounded versus simply being attached to a building.
25. Using the symmetry of the arrangement, show that the net Coulomb force on the charge \(q\) at the center of the square below (Figure) is zero if the charges on the four corners are exactly equal.
Four point charges \(q_{a}\), \(q_{b}\), \(q_{c}\), and \(q_{d}\) lie on the corners of a square and \(q\) is located at its center.
(a) Using the symmetry of the arrangement, show that the electric field at the center of the square in Figure zero if the charges on the four corners are exactly equal.
(b) Show that this is also true for any combination of charges in which \(q_{a}=q_{d}\) and \(q_{b}=q_{c}\)
26. (a) What is the direction of the total Coulomb force on \(q\) in Figure if \(q\) is negative, \(q_{a}= q_{c}\) and both are negative, and \(q_{b}=q_{c}\) and both are positive? (b) What is the direction of the electric field at the center of the square in this situation?
27. Considering Figure, suppose that \(q_{a}=q_{d}\) and \(q_{b}=q_{c}\). First show that \(q\) is in static equilibrium. (You may neglect the gravitational force.) Then discuss whether the equilibrium is stable or unstable, noting that this may depend on the signs of the charges and the direction of displacement of \(q\) from the center of the square.
28. If \(q_{a}=0\) in Figure, under what conditions will there be no net Coulomb force on \(q\)?
29. In regions of low humidity, one develops a special “grip” when opening car doors, or touching metal door knobs. This involves placing as much of the hand on the device as possible, not just the ends of one’s fingers. Discuss the induced charge and explain why this is done.
30. Tollbooth stations on roadways and bridges usually have a piece of wire stuck in the pavement before them that will touch a car as it approaches. Why is this done?
31. Suppose a woman carries an excess charge. To maintain her charged status can she be standing on ground wearing just any pair of shoes? How would you discharge her? What are the consequences if she simply walks away?
Problems & Exercises
18.1: Static Electricity and Charge: Conservation of Charge
32. Common static electricity involves charges ranging from nanocoulombs to microcoulombs.
(a) How many electrons are needed to form a charge of \(–2.00nC\)
(b) How many electrons must be removed from a neutral object to leave a net charge of \(0.500\mu C\) ?
Solution
a. \(1.25\times 10^{10}\)
b. \(3.13\times 10^{12}\)
33. If \(1.80\times 10^{20}\) electrons move through a pocket calculator during a full day’s operation, how many coulombs of charge moved through it?
34. To start a car engine, the car battery moves \(3.75\times 10^{21}\) electrons through the starter motor. How many coulombs of charge were moved?
Solution
-600C
35. A certain lightning bolt moves 40.0 C of charge. How many fundamental units of charge \(|q_{e}|\) is this?
18.2: Conductors and Insulators
36. Suppose a speck of dust in an electrostatic precipitator has \(1.0000\times 10^{12}\) protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
Solution
\(1.03\times 10^{12}\)
37. An amoeba has \(1.00\times 10^{16}\) protons and a net charge of 0.300 pC. (a) How many fewer electrons are there than protons? (b) If you paired them up, what fraction of the protons would have no electrons?
38. A 50.0 g ball of copper has a net charge of \(2.00\mu C\). What fraction of the copper’s electrons has been removed? (Each copper atom has 29 protons, and copper has an atomic mass of 63.5.)
Solution
\(9.09\times 10^{-13}\)
39. What net charge would you place on a 100 g piece of sulfur if you put an extra electron on 1 in \(10^{12}\) of its atoms? (Sulfur has an atomic mass of 32.1.)
40. How many coulombs of positive charge are there in 4.00 kg of plutonium, given its atomic mass is 244 and that each plutonium atom has 94 protons?
Solution
\(1.48\times 10^{8}C\)
18.3: Coulomb's Law
41. What is the repulsive force between two pith balls that are 8.00 cm apart and have equal charges of – 30.0 nC?
42. (a) How strong is the attractive force between a glass rod with a \(0.700\mu C\) charge and a silk cloth with a \(-0.600\mu C\) charge, which are 12.0 cm apart, using the approximation that they act like point charges?
(b) Discuss how the answer to this problem might be affected if the charges are distributed over some area and do not act like point charges.
Solution
(a) 0.263 N
(b) If the charges are distributed over some area, there will be a concentration of charge along the side closest to the oppositely charged object. This effect will increase the net force.
43. Two point charges exert a 5.00 N force on each other. What will the force become if the distance between them is increased by a factor of three?
44. Two point charges are brought closer together, increasing the force between them by a factor of 25. By what factor was their separation decreased?
Solution
The separation decreased by a factor of 5.
45. How far apart must two point charges of 75.0 nC (typical of static electricity) be to have a force of 1.00 N between them?
46. If two equal charges each of 1 C each are separated in air by a distance of 1 km, what is the magnitude of the force acting between them? You will see that even at a distance as large as 1 km, the repulsive force is substantial because 1 C is a very significant amount of charge.
47. A test charge of \(+2\mu C\) is placed halfway between a charge of \(+6\mu C\) and another of \(+4\mu C\) separated by 10 cm.
(a) What is the magnitude of the force on the test charge?
(b) What is the direction of this force (away from or toward the \(+6\mu C\) charge)?
48. Bare free charges do not remain stationary when close together. To illustrate this, calculate the acceleration of two isolated protons separated by 2.00 nm (a typical distance between gas atoms). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.
Solution
\[F=k\dfrac{|q_{1}q_{2}|}{r^{2}}=ma\Rightarrow a=\dfrac{kq^{2}}{mr^{2}}\]
\[=\dfrac{(9.00\times 10^{9}N\cdot m^{2}/C^{2})(1.60\times 10^{-19}m)^{2}}{(1.67\times 10^{-27}kg)(2.00\times 10^{-9}m)^{2}}\]
\[=3.45\times 10^{16}m/s^{2}\]
49. (a) By what factor must you change the distance between two point charges to change the force between them by a factor of 10?
(b) Explain how the distance can either increase or decrease by this factor and still cause a factor of 10 change in the force.
Solution
(a) 3.2
(b) If the distance increases by 3.2, then the force will decrease by a factor of 10 ; if the distance decreases by 3.2, then the force will increase by a factor of 10. Either way, the force changes by a factor of 10.
50. Suppose you have a total charge \(q_{tot}\) that you can split in any manner. Once split, the separation distance is fixed. How do you split the charge to achieve the greatest force?
51. (a) Common transparent tape becomes charged when pulled from a dispenser. If one piece is placed above another, the repulsive force can be great enough to support the top piece’s weight. Assuming equal point charges (only an approximation), calculate the magnitude of the charge if electrostatic force is great enough to support the weight of a 10.0 mg piece of tape held 1.00 cm above another.
(b) Discuss whether the magnitude of this charge is consistent with what is typical of static electricity.
Solution
(a) \(1.04\times 10^{-9}C\)
(b) This charge is approximately 1 nC, which is consistent with the magnitude of charge typical for static electricity
52. (a) Find the ratio of the electrostatic to gravitational force between two electrons.
(b) What is this ratio for two protons?
(c) Why is the ratio different for electrons and protons?
53. At what distance is the electrostatic force between two protons equal to the weight of one proton?
54. A certain five cent coin contains 5.00 g of nickel. What fraction of the nickel atoms’ electrons, removed and placed 1.00 m above it, would support the weight of this coin? The atomic mass of nickel is 58.7, and each nickel atom contains 28 electrons and 28 protons.
Solution
\(1.02\times 10^{-11}\)
55. (a) Two point charges totaling \(8.00 \mu C\) exert a repulsive force of 0.150 N on one another when separated by 0.500 m. What is the charge on each?
(b) What is the charge on each if the force is attractive?
56. Point charges of \(5.00 \mu C\) and \(-3.00 \mu C\) are placed 0.250 m apart.
(a) Where can a third charge be placed so that the net force on it is zero?
(b) What if both charges are positive?
Solution
(a). 0.859 m beyond negative charge on line connecting two charges
(b). 0.109 m from lesser charge on line connecting two charges
57. Two point charges \(q_{1}\) and \(q_{2}\) are \(3.00m\) apart, and their total charge is \(20\mu C\).
(a) If the force of repulsion between them is 0.075N, what are magnitudes of the two charges?
(b) If one charge attracts the other with a force of 0.525N, what are the magnitudes of the two charges? Note that you may need to solve a quadratic equation to reach your answer.
18.4: Electric Field: Concept of a Field Revisited
58. What is the magnitude and direction of an electric field that exerts a \(2.00\times 10^{-5} N\) upward force on a \(-1.75 \mu C\) charge?
59. What is the magnitude and direction of the force exerted on a \(3.50\mu C\) charge by a 250 N/C electric field that points due east?
Solution
\(8.75\times 10^{-4}\)
60. Calculate the magnitude of the electric field 2.00 m from a point charge of 5.00 mC (such as found on the terminal of a Van de Graaff).
61. (a) What magnitude point charge creates a 10,000 N/C electric field at a distance of 0.250 m?
(b) How large is the field at 10.0 m?
Solution
(a) \(6.94\times 10^{-8}C\)
(b) \(6.25N/C\)
62. Calculate the initial (from rest) acceleration of a proton in a \(5.00\times 10^{6} N/C\) electric field (such as created by a research Van de Graaff). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics.
63. (a) Find the direction and magnitude of an electric field that exerts a \(4.80\times 10^{-17}N\)westward force on an electron.
(b) What magnitude and direction force does this field exert on a proton?
Solution
(a) 300N/C (east)
(b) \(4.80\times 10^{-17}N\) (east)
18.5: Electric Field Lines: Multiple Charges
64. (a) Sketch the electric field lines near a point charge \(+q\).
(b) Do the same for a point charge \(-3.00q\).
65. Sketch the electric field lines a long distance from the charge distributions shown in Figure (a) and (b)
66. Figure shows the electric field lines near two charges \(q_1\) and \(q_2\). What is the ratio of their magnitudes?
(b) Sketch the electric field lines a long distance from the charges shown in the figure.
The electric field near two charges.
67. Sketch the electric field lines in the vicinity of two opposite charges, where the negative charge is three times greater in magnitude than the positive. (See Figure 18.6.8 for a similar situation).
18.7: Conductors and Electric Fields in Static Equilibrium
68. Sketch the electric field lines in the vicinity of the conductor in Figure given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
69. Sketch the electric field lines in the vicinity of the conductor in Figure 18.8.12 given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
70. Sketch the electric field between the two conducting plates shown in Figure, given the top plate is positive and an equal amount of negative charge is on the bottom plate. Be certain to indicate the distribution of charge on the plates.
71. Sketch the electric field lines in the vicinity of the charged insulator in Figure noting its nonuniform charge distribution.
A charged insulating rod such as might be used in a classroom demonstration.
72. What is the force on the charge located at \(x=8.00cm\) in Figure (a) given that \(q=1.00 \mu C\)?
(a) Point charges located at 3.00, 8.00, and 11.0 cm along the x-axis. (b) Point charges located at 1.00, 5.00, 8.00, and 14.0 cm along the x-axis.
73. (a) Find the total electric field at \(x=1.00cm\) in Figure
(b) given that \(q=5.00nC\). (b) Find the total electric field at \(x=1.00cm\) in Figure 18.8.15(b).
(c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, will there be a single charge, double charge, etc., and what will its value(s) be?)
Solution
(a) \(E_{x=1.00cm}=-\infty\)
(b) \(2.12\times 10^{5} N/C\)
(c) one charge of \(+q\)
74. (a) Find the electric field at \(x=5.00com\) in Figure 18.8.15(a), given that \(q=1.00\mu C\).
(b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for \(-2q\) alone?
(c) Can the electric field be zero anywhere between 0.00 and 8.00 cm?
(d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidly approach zero and why?
(e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity? (Hint: A graphing calculator can yield considerable insight in this problem.)
75. (a) Find the total Coulomb force on a charge of 2.00 nC located at \(x=4.00cm\) in Figure
(b), given that \(q=1.00\mu C\). (b) Find the x -position at which the electric field is zero in Fgiure 18.8.15(b).
Solution
(a) 0.252 N to the left
(b) \(x=6.07cm\)
76. Using the symmetry of the arrangement, determine the direction of the force on \(q\) in the figure below, given that \(q_{a}=q_{b}=+7.50 \mu C\) and \(q_{c}=q_{d}=-7.50 \mu C\).
(b) Calculate the magnitude of the force on the charge \(q\), given that the square is 10.0 cm on a side and \(q=2.00 \mu C\).
77. (a) Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square in Figure 18.8.16, given that \(q_{a}=q_{b}=-1.00 \mu C\) and \(q_{c}=q_{d}=+1.00 \mu C\).
(b) Calculate the magnitude of the electric field at the location of \(q\), given that the square is 5.00 cm on a side.
Solution
(a) The electric field at the center of the square will be straight up, since \(q_{a}\) and \(q_{b}\) are negative and \(q_{c}\) and \(q_{d}\) are positive and all have the same magnitude.
(b) \(2.04\times 10^{7} N/C\) (upward)
78. Find the electric field at the location of \(q_{a}\) in Figure 18.8.16 given that \(q_{b}=q_{c}=q_{d}=+2.00nC\), \(q=-1.00nC\), and the square is 20.0 cm on a side.
79. Find the total Coulomb force on the charge \(q\) in Figure 18.8.16, given that \(q=1.00\mu C\), \(q_{a}=2.00\mu C\), \(q_{b}=-3.00 \mu C\), \(q_{c}=-4.00\mu C\), and \(q_{d}=+1.00 \mu C\). The square is 50.0 cm on a side.
Solution
0.102 N, in the \(-y\) direction
80. (a) Find the electric field at the location of \(q_{a}\) in Figure, given that \(q_{b}=+10.00 \mu C\) and \(q_{c}=-5.00\mu C\).
(b) What is the force on \(q_{a}\), given that \(q_{a}=+1.50nC\)?
Point charges located at the corners of an equilateral triangle 25.0 cm on a side.
81. (a) Find the electric field at the center of the triangular configuration of charges in Figure 18.8.17, given that \(q_{a}=+2.50nC\), \(q_{b}=-8.00nC\), and \(q_{c}=+1.50nC\).
(b) Is there any combination of charges, other than \(q_{a}=q_{b}=q_{c}\), that will produce a zero strength electric field at the center of the triangular configuration?
Solution
(a) \(\overrightarrow{E}=4.36\times 10^{3} N/C, 35.0^{\circ}\), below the horizontal.
(b) No
18.8: Applications of Electrostatics
82. (a) What is the electric field 5.00 m from the center of the terminal of a Van de Graaff with a 3.00 mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal?
(b) At this distance, what force does the field exert on a \(2.00 \mu C\) charge on the Van de Graaff’s belt?
83. (a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth?
(b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.
Solution
(a) \(5.58\times 10^{-11} N/C\)
(b) the coulomb force is extraordinarily stronger than gravity
84. A simple and common technique for accelerating electrons is shown in Figure, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving. (a) Calculate the acceleration of the electron if the field strength is \(2.50 \times 10^[4}N/C\).
(b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole.
Parallel conducting plates with opposite charges on them create a relatively uniform electric field used to accelerate electrons to the right. Those that go through the hole can be used to make a TV or computer screen glow or to produce X-rays.
85. Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface.
(a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center?
(b) What acceleration will the field produce on a free electron near Earth’s surface?
(c) What mass object with a single extra electron will have its weight supported by this field?
Solution
(a) \(-6.76\times 10^{5}C\)
(b) \(2.63\times 10^{13} m/s^{2}\) (upward)
(c) \(2.45\times 10^{-18}kg\)
86. Point charges of \(25.0 \mu C\) and \(45.0 \mu C\) are placed 0.500 m apart.
(a) At what point along the line between them is the electric field zero?
(b) What is the electric field halfway between them?
87. What can you say about two charges \(q_{1}\) and \(q_{2}\), if the electric field one-fourth of the way from \(q_{1}\) to \(q_{2}\) is zero?
Solution
The charge \(q_{2}\) is 9 times greater than \(q_{1}\).
88. Calculate the angular velocity \(\omega\) of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is \(0.530\times 10^{-10}m\). You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction.
89. An electron has an initial velocity of \(5.00\times 10^{6}m/s\) in a uniform \(2.00\times 10^{5} N/C\) strength electric field. The field accelerates the electron in the direction opposite to its initial velocity.
(a) What is the direction of the electric field?
(b) How far does the electron travel before coming to rest?
(c) How long does it take the electron to come to rest? (d) What is the electron’s velocity when it returns to its starting point?
90. The practical limit to an electric field in air is about \(3.00\times 10^{6}N/C\). Above this strength, sparking takes place because air begins to ionize and charges flow, reducing the field.
(a) Calculate the distance a free proton must travel in this field to reach\(3.00%\) of the speed of light, starting from rest.
(b) Is this practical in air, or must it occur in a vacuum?
91. A 5.00 g charged insulating ball hangs on a 30.0 cm long string in a uniform horizontal electric field as shown in Figure. Given the charge on the ball is \(1.00 \mu C\), find the strength of the field.
A horizontal electric field causes the charged ball to hang at an angle of \(8.00^{\circ}\).
92. Figure shows an electron passing between two charged metal plates that create an 100 N/C vertical electric field perpendicular to the electron’s original horizontal velocity. (These can be used to change the electron’s direction, such as in an oscilloscope.) The initial speed of the electron is \(3.00\times 10^{6}m/s\), and the horizontal distance it travels in the uniform field is 4.00 cm.
(a) What is its vertical deflection?
(b) What is the vertical component of its final velocity?
(c) At what angle does it exit? Neglect any edge effects.
93. The classic Millikan oil drop experiment was the first to obtain an accurate measurement of the charge on an electron. In it, oil drops were suspended against the gravitational force by a vertical electric field. (Figure 18.9.9) Given the oil drop to be \(1.00\mu m\) in radius and have a density of \(920 kg/m^{3}\): (a) Find the weight of the drop. (b) If the drop has a single excess electron, find the electric field strength needed to balance its weight.
In the Millikan oil drop experiment, small drops can be suspended in an electric field by the force exerted on a single excess electron. Classically, this experiment was used to determine the electron charge \(q_{e}\) by measuring the electric field and mass of the drop.
94. (a) In Figure, four equal charges \(q\) lie on the corners of a square. A fifth charge \(Q\) is on a mass \(m\) directly above the center of the square, at a height equal to the length \(d\) of one side of the square. Determine the magnitude of \(q\) in terms of \(Q\),\(m\), and \(d\), if the Coulomb force is to equal the weight of \(m\). (b) Is this equilibrium stable or unstable? Discuss.
Four equal charges on the corners of a horizontal square support the weight of a fifth charge located directly above the center of the square.
95. (a) Calculate the electric field strength near a 10.0 cm diameter conducting sphere that has 1.00 C of excess charge on it.
(b) What is unreasonable about this result? (c) Which assumptions are responsible?
96. (a) Two 0.500 g raindrops in a thunderhead are 1.00 cm apart when they each acquire 1.00 mC charges. Find their acceleration.
(b) What is unreasonable about this result?
(c) Which premise or assumption is responsible?
97.  If a car has a 1000 kg mass and the ball is to be able to lift it from a distance of 1.00 m:
(a) What minimum charge must be used?
(b) What is the electric field near the surface of the ball?
(c) Why are these results unreasonable?
(d) Which premise or assumption is responsible?
98. Consider two insulating balls with evenly distributed equal and opposite charges on their surfaces, held with a certain distance between the centers of the balls.  Choose interesting points and comment on the meaning of the field at those points. For example, at what points might the field be just that due to one ball and where does the field become negligibly small? Among the things to be considered are the magnitudes of the charges and the distance between the centers of the balls. Your instructor may wish for you to consider the electric field off axis or for a more complex array of charges, such as those in a water molecule.
99. Consider identical spherical conducting space ships in deep space where gravitational fields from other bodies are negligible compared to the gravitational attraction between the ships. Construct a problem in which you place identical excess charges on the space ships to exactly counter their gravitational attraction. Calculate the amount of excess charge needed. Examine whether that charge depends on the distance between the centers of the ships, the masses of the ships, or any other factors. Discuss whether this would be an easy, difficult, or even impossible thing to do in practice.
Contributors and Attributions
-
Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


3.6: History of Business Ethics
Learning Objectives
- Discuss the history of business ethics
What comes to mind when you hear the phrase “business ethics?” Many people associate ethics with following your conscience or the law. However, business ethics is a lot more complicated. Business ethics is not simply following the law or choosing to follow along with the status quo of other companies. Instead, business ethics are principles and values that guide how a company does business. Like organizational behavior, there is no exact time and place in which business ethics were created. However, there are historical events that have helped to develop each aspect of the term. Let’s explore some of these events and how they contributed to business ethics as we know them today.
Richard T. De George describes the history of business ethics to have blended together from different strands. He believes each strand represents a different yet equally important perspective on business ethics. Depending on what perspective historians have on the topic, will shift their version of history. [1] For the purpose of this section, let’s examine a few of the strands he has identified to ensure we get more of a “whole picture” perspective.
The first strand De George discusses is the one that can be dated back the furthest. “Ethics in business” as he refers to it, includes the history of moral standards used in business practices since the dawn of time. Instead of having a code of ethics like many companies do today, people centuries ago would still use ethical guidelines like the Ten Commandments or Aristotle’s economic relations philosophies. Even after the fall of Rome, clear discussions of business ethics did not exist, however, there were many conversations about justly and honestly buying and selling good and services. Religious organizations used morals and biblical teachings to support ethical behavior, however, it was not until the mid 1900s that the term “ethics” or “ethics in business” was coined.
The next, and more recent strand De George investigates is the academic field of business ethics. During Vietnam, Americans became hypercritical of ongoing government and business operations. Big businesses were booming and the chemical industry was growing rapidly. These swift changes in growth drew a lot of attention and protest. People were concerned for equal employment opportunities and upset about the environmental damage caused by large corporations and the chemical industry. To help counter the protests, some corporations began to focus on a new term, social responsibility. Social responsibility was a term used by companies to help correct the error in their ways. If they were being accused of hiring discrimination, companies could then tell the public that they had a social responsibility to hire a diverse group of individuals. A company was accused of too much pollution? They can use social responsibility to change their impact on the environment and promote cutting back on harmful pollutants. Social responsibility differed from one industry to another since there were no set standards or descriptors in place to clearly define it. In the 1960s, social responsibility entered into the world of academia and is still a popular class in many business schools today. However, it wasn’t until the 1970s that business ethics as an academic field of study was born.
In the 1970s, scholars, theologians, and philosophers were all investigating the world of business. With the onset of so many new perspectives and ideas, the term business ethics was born. In November 1974, the first ever business ethics conference was held at the University of Kansas. The ideas and case studies discussed at the conference could be found in new business courses in the years to follow. Professors of philosophy and business alike were working together to fully examine and understand this new field. So what was the difference between social responsibility and business ethics? Business ethics were developed with a clearer set of boundaries and expectations. In addition, business ethics created more controversy within the business world than social responsibility did. Business ethics took a more critical approach to how companies operated and many argued that ethical decisions were outside of a company’s control. As we continue through this module, you will learn how companies actually can enforce business ethics and why it is so important and impactful to the success of their company.
Practice Question
-
De George, Richard T. "A History of Business Ethics." Markkula Center for Applied Ethics. Accessed April 10, 2019.
https://www.scu.edu/ethics/focus-areas/business-ethics/resources/a-history-of-business-ethics/
. ↵
Contributors and Attributions
- History of Business Ethics. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Aristotle, Ethica Nicomachea - page 1 from 1837 - {book-scan}. Authored by : Karl-Ludwig Poggemann. Provided by : Flickr. Located at : https://flic.kr/p/8LnqMT . License : CC BY: Attribution


1.3.1: Integer Exponents: a Review with Variables
By the end of this section, you will be able to:
- Simplify expressions using the properties for exponents
- Use the definition of a negative exponent
Before you get started, take this readiness quiz.
1. Simplify \((−2)(−2)(−2)\).
2. Simplify \((2x^5)^3\).
3. Simplify \((2x^2)(4x^3)\).
A Review of Positive Integer Exponents
Remember that a positive integer exponent indicates repeated multiplication of the same quantity. For example, in the expression \(a^m\), the positive integer exponent \(m\) tells us how many times we use the base \(a\) as a factor.
\[a^{m}= \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{\text{m factors}} \nonumber\]
For example,
\[(-9)^{5}= \underbrace{ (-9)\cdot (-9)\cdot (-9)\cdot (-9) \cdot (-9)}_{\text{5 factors}} \nonumber\]
Let’s review the vocabulary for expressions with exponents.
\(a^m = \underbrace{a\cdot a\cdot a \cdots a}_{m \text{ factors }}\)
This is read \(a\) to the \(m^{th}\) power .
In the expression \(a^m\) with positive integer \(m\) and \(a\not=0\), the exponent \(m\) tells us how many times we use the base \(a\) as a factor.
Recall the following example which leads to the the Product Property for Positive Integer Exponents .
|
\(\quad x^{2} x^{3}\) |
|
|---|---|
| What does this mean? |
\(=\underbrace{x \cdot x}_{2 \text{ factors}\space} \cdot \underbrace{ x\cdot x\cdot x}_{3 \text{ factors}}\) \(=\underbrace{x\cdot x\cdot x\cdot x\cdot x}_{5 \text{ factors}}\) |
| \(= x^{5}\) |
The base stayed the same and we added the exponents.
If \(a\) is a real number and \(m\) and \(n\) are positive integers, then
\[a^ma^n=a^{m+n}. \nonumber\]
To multiply with like bases, add the exponents.
Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.
| Consider | \(\quad\dfrac{x^5}{x^2}\) | and | \(\quad\dfrac{x^2}{x^3}\) |
|---|---|---|---|
| What do they mean? | \(=\dfrac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\) | \(=\dfrac{x\cdot x}{x\cdot x\cdot x}\) | |
| Use the Equivalent Fractions Property. | \(=\dfrac{\cancel{x}\cdot\cancel{x}\cdot x\cdot x\cdot x}{\cancel{x}\cdot \cancel{x}}\) | \(=\dfrac{\cancel{x}\cdot \cancel{x}\cdot 1}{\cancel{x}\cdot \cancel{x}\cdot x}\) | |
| Simplify. | \(=x^3\) |
\(=\dfrac{1}{x}\) |
|
| Note. | \(x^3=x^{5-2}\) | \(\dfrac{1}{x}=\dfrac{1}{x^1}=\dfrac{1}{x^{3-2}}\) |
Here we see how to use the initial exponents to arrive at the simplified expression. When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator--notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator: \(\dfrac{\cancel{x}}{\cancel{x}}=1\). This leads to the Quotient Property for Positive Integer Exponents .
If \(a\) is a real number, \(a \neq 0\), and \(m\) and \(n\) are distinct positive integers, then
\[\dfrac{a^m}{a^n}=\begin{cases}a^{m−n}& \text{ if } m>n \\ \dfrac{1}{a^{n−m}} & \text{ if } n>m. \end{cases}\nonumber\]
Simplify each expression:
a. \(\dfrac{x^9}{x^7}\)
b. \(\dfrac{3^{10}}{3^2}\)
c. \(\dfrac{b^8}{b^{12}}\)
d. \(\dfrac{7^3}{7^5}\)
Solution
-
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
a.
\(\dfrac{x^9}{x^7}\) Since \(9>7\), there are more factors of \(x\) in the numerator.
Use the Quotient Property, \(\dfrac{a^m}{a^n}=a^{m−n}\), for \(m>n\).
\(=x^{9-7}\) Simplify. \(=x^2\) Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
b.
\(\quad\dfrac{3^{10}}{3^2}\) Since \(10>2\), there are more factors of \(3\) in the numerator.
Use the Quotient Property, \(\dfrac{a^m}{a^n}=a^{m−n}\), for \(m>n\).
\(=3^{10-2}\) Simplify. \(=3^8\) Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
c.
\(\quad\dfrac{b^8}{b^{12}}\) Since \(12>8\), there are more factors of \(b\) in the denominator.
Use the Quotient Property, \(\dfrac{a^m}{a^n}=\dfrac{1}{a^{n−m}}\), for \(n>m\).
\(=\dfrac{1}{b^{12-8}}\) Simplify. \(=\dfrac{1}{b^4}\) Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
d.
\(\quad \dfrac{7^3}{7^5}\) Since \(5>3\), there are more factors of \(3\) in the denominator.
Use the Quotient Property, \(\dfrac{a^m}{a^n}=\dfrac{1}{a^{n−m}}\) for \(n>m\).
\(=\dfrac{1}{7^{5-3}}\) Simplify. \(=\dfrac{1}{7^2}\) Simplify. \(=\dfrac{1}{49}\) -
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Simplify each expression:
a. \(\dfrac{x^{15}}{x^{10}}\)
b. \(\dfrac{6^{14}}{6^5}\)
c. \(\dfrac{x^{18}}{x^{22}}\)
d. \(\dfrac{12^{15}}{12^{30}}\)
- Answer
-
a. \(x^5\)
b. \(6^9\)
c. \(\dfrac{1}{x^4}\)
d. \(\dfrac{1}{12^{15}}\)
Simplify each expression:
a. \(\dfrac{y^{43}}{y^{37}}\)
b. \(\dfrac{10^{15}}{10^{7}}\)
c. \(\dfrac{m^7}{m^{15}}\)
d. \(\dfrac{9^8}{9^{19}}\)
- Answer
-
a. \(y^6\)
b. \(10^8\)
c. \(\dfrac{1}{m^8}\)
d. \(\dfrac{1}{9^{11}}\)
Extending the Meaning of Exponent to Integers
Note that, while so far an exponent that is not a positive integer has no meaning, we see that blindly applying the above properties for such exponents leads to a couple definitions.
| \(\quad 1\) | |
|---|---|
|
For \(m\) a positive integer (so that \(x^m\) has meaning, \(x\neq 0\)). |
\(=\dfrac{x^m}{ x^m}\) |
|
Blindly follow the quotient rule that we know to be true in another case. |
\(=x^{m-m}\) |
| Reduce the fraction on the left. | \(=x^0\) |
| Conclude. |
\(x^0=1\) |
So if the Quotient Property is also to hold for the exponent zero we must define, for \(x\not= 0\),
\[x^0=1. \nonumber \]
Similarly, consider this expression where \(m\) is a positive integer (so that \(x^m\) has meaning).
| \(\quad x^{-m}x^m\) | |
|---|---|
| Blindly applying the product property. | \(=x^{-m+m}\) |
| Simplify exponent. | \(=x^0\) |
| Using our new definition: \(x^0=1\). | \(=1\) |
| Draw conclusion. | \(x^{-m}x^m=1\) |
| Note the property of the reciprocal. | \(x^{-m}\) is the reciprocal of \(x^m\). |
| Rewrite in symbols. | \(x^{-m}=\dfrac{1}{x^m}\) |
So, if the product property of exponents holds also for (so far undefined) negative integers, we must define, for \(m\) a negative integer,
\[x^{-m}=\dfrac{1}{x^m}.\nonumber\] Also, since taking the reciprocal of both sides preserves the equality we also have, equivalently,
\[\dfrac{1}{x^{-m}}=x^m.\nonumber\]
We could also write these two statements above simultaneously as \[x^{-m}=\dfrac{1}{x^m}, m\text{ any integer.}\nonumber\]
So, we must define
For \(a\) any non-zero real number
\[a^0=1\nonumber\]
and for \(m\) any positive integer
\[a^{-m}=\dfrac{1}{a^m} \text{ or, equivalently, }\dfrac{1}{a^{-m}}=a^m.\nonumber\]
In the above, the base can be anything (\(x\) can be anything) which we know to be different from zero, and in this text, we assume any variable that we raise to the zero power is not zero.
Simplify each expression:
a. \(9^0\)
b. \(n^0\)
c. \((−4a^2b)^0\)
d. \(-3^0\)
Solution
The definition says any non-zero number raised to the zero power is \(1\).
a. Use the definition of the zero exponent. \(9^0 = 1\)
b.
Use the definition of the zero exponent. \(n^0 = 1\)
To simplify the expression \(n\) raised to the zero power we just use the definition of the zero exponent. The result is \(1\).
c. Anything raised to the power zero is 1. Here the base is \(-4a^2 b\), so \((−4a^2b)^0=1\)
d. Anything raised to the power zero is 1. Here the base is 3, so this is the opposite of \(3^0\), or, the opposite of 1. So, \(-3^0=-1\)
Simplify each expression:
a. \(11^0\)
b. \(q^0\)
c. \((−12p^3q^2)^0\)
d. \(-7^0\)
- Answer
-
a. 1
b. 1
c. 1
d. -1
Simplify each expression:
a. \(23^0\)
b. \(r^0\)
c. \((2s t^5)^0\)
d. \(-s^0\)
- Answer
-
a. 1
b. 1
c. 1
d. -1
Simplify each expression. Write your answer using positive exponents.
a. \(x^{−5}\)
b. \(10^{−3}\)
c. \(\dfrac{1}{y^{−4}}\)
d. \(\dfrac{1}{3^{−2}}\)
Solution
a.
| \(\quad x^{−5}\) | |
|---|---|
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{1}{x^5}\) |
b.
| \(\quad 10^{−3}\) | |
|---|---|
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{1}{10^3}\) |
| Simplify. | \(=\dfrac{1}{1000}\) |
c.
| \(\quad \dfrac{1}{y^{-4}}\) | |
|---|---|
| Use the definition of a negative exponent, \(\dfrac{1}{a^{−n}}=a^n\). | \(=y^4\) |
d.
| \(\quad \dfrac{1}{3^{-2}}\) | |
|---|---|
| Use the definition of a negative exponent, \(\dfrac{1}{a^{−n}}=a^n\). | \(=3^2\) |
| Simplify. | \(=9\) |
Simplify each expression. Write your answer using positive exponents.
a. \(z^{−3}\)
b. \(10^{−7}\)
c. \(\dfrac{1}{p^{−8}}\)
d. \(\dfrac{1}{4^{−3}}\)
- Answer
-
a. \(\dfrac{1}{z^3}\)
b. \(\dfrac{1}{10,000,000}\)
c. \(p^8\)
d. \(64\)
Simplify each expression. Write your answer using positive exponents.
a. \(n^{−2}\)
b. \(10^{−4}\)
c. \(\dfrac{1}{q^{−7}}\)
d. \(\dfrac{1}{2^{−4}}\)
- Answer
-
a. \(\dfrac{1}{n^2}\)
b. \(\dfrac{1}{10,000}\)
c. \(q^7\)
d. \(16\)
Properties of Negative Exponents
The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.
Any expression that has negative exponents is not considered to be in simplest form . We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.
For example, if after simplifying an expression we end up with the expression \(x^{−3}\), we will take one more step and write \(\dfrac{1}{x^3}\). The answer is considered to be in simplest form when it has only positive exponents.
Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.
| \(\quad\left( \dfrac{3}{4} \right)^{-2}\) | |
|---|---|
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{1}{\left( \dfrac{3}{4} \right)^{2}}\) |
| Simplify the denominator. | \(=\dfrac{1}{\dfrac{9}{16}}\) |
| Simplify the complex fraction. | \(=\dfrac{16}{9}\) |
| But we know that | \(\dfrac{16}{9}=\left( \dfrac{4}{3} \right)^{2}\) |
| This tells us that | \(\left( \dfrac{3}{4} \right)^{-2} = \left( \dfrac{4}{3} \right)^{2}\) |
To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.
This leads us to the Quotient to a Negative Integer Exponent Property .
If \(a\) and \(b\) are real numbers, \(a\neq 0\), \(b\neq 0\), and \(n\) is an integer, then
\[\left(\dfrac{a}{b}\right)^{−n}=\left(\dfrac{b}{a}\right)^n. \nonumber \]
Simplify each expression. Write your answer using positive exponents.
a. \(\left( \dfrac{5}{7} \right)^{−2}\)
b. \(\left( −\dfrac{x}{y} \right)^{−3}\)
Solution
a.
| \(\quad \left( \dfrac{5}{7} \right)^{−2}\) | |
|---|---|
| Use the Quotient to a Negative Integer Exponent Property, \(\left(\dfrac{a}{b} \right)^{−n}= \left( \dfrac{b}{a} \right)^n\). | |
| Take the reciprocal of the fraction and change the sign of the exponent. | \(=\left( \dfrac{7}{5}\right)^2\) |
| Simplify. | \(=\dfrac{49}{25}\) |
b.
| \(\quad\left( −\dfrac{x}{y} \right)^{−3}\) | |
|---|---|
| Use the Quotient to a Negative Integer Exponent Property, \(\left(\dfrac{a}{b} \right)^{−n}= \left( \dfrac{b}{a} \right)^n\). | |
| Take the reciprocal of the fraction and change the sign of the exponent. | \(=\left( -\dfrac{y}{x}\right)^3\) |
| Simplify. | \(=-\dfrac{y^3}{x^3}\) |
Simplify each expression. Write your answer using positive exponents.
a. \(\left(\dfrac{2}{3}\right)^{−4}\)
b. \(\left(−\dfrac{m}{n}\right)^{−2}\)
- Answer
-
a. \(\dfrac{81}{16}\)
b. \(\dfrac{n^2}{m^2}\)
Simplify each expression. Write your answer using positive exponents.
a. \(\left(\dfrac{3}{5}\right)^{−3}\)
b. \(\left(−\dfrac{a}{b}\right)^{−4}\)
- Answer
-
a. \(\dfrac{125}{27}\)
b. \(\dfrac{b^4}{a^4}\)
We would like to verify that the properties of positive integer exponents can be extended to all integer exponents. We will do some examples. Let \(m\) and \(n\) be non-negative integers, \(m>n\). Let's simplify \(x^mx^{-n}\).
| \(\quad x^mx^{-n}\) | |
|---|---|
| Use the definition: \(x^{-n}=\dfrac{1}{x^n}\). | \(=x^m\cdot\dfrac{1}{x^n}\) |
| Multiply fractions. | \(=\dfrac{x^m}{x^n}\) |
| Use the Quotient Property of exponents. | \(x^{m-n}\quad\) or \(\quad x^{m+(-n)}\) |
| Conclude. | \(x^mx^{-n}=x^{m-n}=x^{m+(-n)}\) |
So, the product property holds in this case.
And if \(m<n\),
| \(x^mx^{-n}\) | |
|---|---|
| Use the definition: \(x^{-n}=\dfrac{1}{x^n}\). | \(=x^m\cdot\dfrac{1}{x^n}\) |
| Multiply fractions. | \(=\dfrac{x^m}{x^n}\) |
| Use the Quotient Property of exponents. | \(=\dfrac{1}{x^{n-m}}\) |
| Use second variation of the definition of negative exponents. | \(=x^{-(n-m)}\) |
| Simplify. | \(x^{m-n}\quad \) or \(\quad x^{m+(-n)}\) |
| Conclude. | \(x^mx^{-n}=x^{m-n}=x^{m+(-n)}\) |
So the Product Property for Positive Integer Exponents holds in this case, too. We can in a similar way check other combinations to see that the Product Property for Positive Integer Exponents holds for all integers. The Product Property for Integer Exponents follows directly
\[\dfrac{x^m}{x^n}=x^m\cdot \dfrac{1}{x^{n}}=x^m\cdot x^{-n}=x^{m-n}\nonumber\]
and
\[\dfrac{x^m}{x^n}=\dfrac{1}{x^{-m}}\cdot \dfrac{1}{x^{n}}=\dfrac{1}{x^{-m}\cdot x^{n}}=\dfrac{1}{x^{-m+n}}\quad \text{ or }\quad\dfrac{1}{x^{n-m}}.\nonumber\]
So, both the Product Property and the Quotient Property hold for all integer exponents.
If \(a\) is a real number and \(m\) and \(n\) are integers, then
\[a^ma^n=a^{m+n}. \nonumber\]
To multiply with like bases, add the exponents.
If \(a\) is a real number, \(a \neq 0\), and \(m\) and \(n\) are integers, then
\[ \dfrac{a^m}{a^n}=a^{m−n}\nonumber \]
and
\[\dfrac{a^m}{a^n}=\dfrac{1}{a^{n−m}}. \nonumber \]
To divide with like bases, subtract the exponents as above.
We will now use the product property with expressions that have negative exponents. We can choose to use the meaning of negative exponents instead of the quotient property above, so we will focus here on the product property.
Simplify each expression:
a. \(z^{−5}z^{−3}\)
b. \((m^4n^{−3})(m^{−5}n^{−2})\)
c. \((2x^{−6}y^8)(−5x^5y^{−3})\)
Solution
a.
| \(\quad z^{−5}z^{−3}\) | |
|---|---|
| Add the exponents, since the bases are the same. | \(=z^{−5−3}\) |
| Simplify. | \(=z^{−8}\) |
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{1}{z^8}\) |
b.
| \(\quad(m^4n^{−3})(m^{−5}n^{−2})\) | |
|---|---|
| Use the Commutative Property to get like bases together. | \(=m^4m^{−5}n^{−2}n^{−3}\) |
| Add the exponents for each base. | \(=m^{−1}n^{−5}\) |
| Use the definition of a negative exponent, \(a^{-n}= \dfrac{1}{a^n}\). | \(=\dfrac{1}{m^1}\cdot\dfrac{1}{n^5}\) |
| Simplify. | \(=\dfrac{1}{mn^5}\) |
c.
| \(\quad (2x^{−6}y^8)(−5x^5y^{−3})\) | |
|---|---|
| Rewrite with the like bases together. | \(=2(−5)(x^{−6}x^5)(y^8y^{−3})\) |
| Multiply the coefficients and add the exponents of each variable. | \(=−10x^{−1}y^5\) |
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=−10\dfrac{1}{x}y^5\) |
| Simplify. | \(=−\dfrac{10y^5}{x}\) |
Simplify each expression:
a. \(z^{−4}z^{−5}\)
b. \((p^6q^{−2})(p^{−9}q^{−1})\)
c. \((3u^{−5}v^7)(−4u^4v^{−2})\)
- Answer
-
a. \(\dfrac{1}{z^9}\)
b. \(\dfrac{1}{p^3q^3}\)
c. \(−\dfrac{12v^5}{u}\)
Simplify each expression:
a. \(c^{−8}c^{−7}\)
b. \((r^5s^{−3})(r^{−7}s^{−5})\)
c. \((−6c^{−6}d^4)(−5c^{−2}d^{−1})\)
- Answer
-
a. \(\dfrac{1}{c^{15}}\)
b. \(\dfrac{1}{r^2s^8}\)
c. \(\dfrac{30d^3}{c^8}\)
Now let’s look at an exponential expression that contains a power raised to a power. Let's see if we can discover a general property.
| \(\quad (x^2)^3\) | |
|---|---|
| What does this mean? | \(=x^2x^2x^2\) |
| How many factors altogether? |
\(=\underbrace{x\cdot x}_{2\; factors }\cdot \underbrace{x\cdot x}_{2\text{ factors }}\cdot \underbrace{x\cdot x}_{2\text{ factors }}\) \(=\underbrace{x\cdot x\cdot x\cdot x \cdot x\cdot x}_{6\text{ factors }}\) |
| So we have | \(=x^6\) |
Notice the 6 is the product of the exponents, 2 and 3. We see that \((x^2)^3\) is \(x^{2\cdot 3}\) or \(x^6\). We can also see that
\[(x^{-2})^3=\left(\dfrac{1}{x^2}\right)^3=\dfrac{1}{x^2}\cdot\dfrac{1}{x^2}\cdot\dfrac{1}{x^2}=\dfrac{1}{x^6}=x^{-6}\nonumber\] so that \((x^{-2})^3=x^{-6}\). In these examples we multiplied the exponents.
We can check various combinations of signs of the exponents which leads us to the Power Property for Integer Exponents.
If \(a\) is a real number and \(m\) and \(n\) are integers, then
\[(a^m)^n=a^{mn}. \nonumber \]
To raise a power to a power, multiply the exponents.
Simplify each expression:
a. \((y^5)^9\)
b. \((4^{-4})^7\)
c. \((y^3)^6(y^5)^4\)
Solution
a.
| \((y^5)^9\) | |
|---|---|
| Use the power property, \((a^m)^n=a^{mn}\). | \(y^{5\cdot 9}\) |
| Simplify. | \(y^{45}\) |
b.
| \(\quad(4^{-4})^7\) | |
|---|---|
| Use the power property. | \(=4^{-4\cdot 7}\) |
| Simplify. | \(=4^{-28}\) |
c.
| \(\quad (y^3)^6(y^5)^4\) | |
|---|---|
| Use the power property. | \(=y^{18}y^{20}\) |
| Add the exponents. | \(=y^{38}\) |
Simplify each expression:
a. \((b^7)^5\)
b. \((5^4)^{-3}\)
c. \((a^4)^5(a^7)^4\)
- Answer
-
a. \(b^{35}\)
b. \(5^{-12}\)
c. \(a^{48}\)
Simplify each expression:
a. \((z^6)^9\)
b. \((3^{-7})^7\)
c. \((q^4)^5(q^3)^3\)
- Answer
-
a. \(z^{54}\)
b. \(3^{-49}\)
c. \(q^{29}\)
We will now look at an expression containing a product that is raised to a power. Can we find this pattern?
| \(\quad(2x)^3\) | |
|---|---|
| What does this mean? | \(=2x\cdot 2x \cdot 2x\) |
| We group the like factors together. | \(=2\cdot 2\cdot 2xxx\) |
| How many factors of \(2\) and of \(x\)? | \(=2^3x^3\) |
Notice that each factor was raised to the power and \((2x)^3\) is \(2^3x^3\).
The exponent applies to each of the factors! We can say that the exponent distributes over multiplication. If we were to check various examples with exponents which are negative or zero, then we would find the same pattern emerges. This leads to the Product to a Power Property for Integer Exponents.
If \(a\) and \(b\) are real numbers and \(m\) is an integer, then
\[(ab)^m=a^mb^m \nonumber. \]
To raise a product to a power, raise each factor to that power.
Simplify each expression:
a. \((−3mn)^3\)
b. \((6k^3)^{−2}\)
c. \((5x^{−3})^2\)
Solution
a.
| \(\quad(−3mn)^3\) | |
|---|---|
| Use Power of a Product Property, \((ab)^m=a^mb^m\). | \(=(−3)^3m^3n^3\) |
| Simplify. | \(=−27m^3n^3\) |
b.
| \(\quad (6k^3)^{−2}\) | |
|---|---|
| Use the Power of a Product Property, \((ab)^m=a^mb^m\). | \(=6^{−2}(k^3)^{−2}\) |
| Use the Power Property, \((a^m)^n=a^{mn}\). | \(=6^{−2}k^{−6}\) |
| Use the definition of a negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{1}{6^2}\cdot\dfrac{1}{k^6}\) |
| Simplify. | \(=\dfrac{1}{36k^6}\) |
c.
| \(\quad(5x^{−3})^2\) | |
|---|---|
| Use the power of a product property, \((ab)^m=a^mb^m\). | \(=5^2(x^{−3})^2\) |
| Simplify. | \(=25x^{−6}\) |
| Rewrite \(x^{−6}\) using \(a^{−n}=\dfrac{1}{a^n}\). | \(=25\dfrac{1}{x^6}\) |
| Simplify. | \(=\dfrac{25}{x^6}\) |
Simplify each expression:
a. \((2wx)^5\)
b. \((2b^3)^{−4}\)
c. \((8a^{−4})^2\)
- Answer
-
a. \(32w^5x^5\)
b. \(\dfrac{1}{16b^{12}}\)
c. \(\dfrac{64}{a^8}\)
Simplify each expression:
a. \((−3y)^3\)
b. \((−4x^4)^{−2}\)
c. \((2c^{−4})^3\)
- Answer
-
a. \(−27y^3\)
b. \(\dfrac{1}{16x^8}\)
c. \(8c^{12}\)
Now we will look at an example that will lead us to the quotient to a power property.
| \(\quad\left( \dfrac{x}{y}\right)^3\) | |
|---|---|
| This means | \(=\dfrac{x}{y}\cdot\dfrac{x}{y}\cdot\dfrac{x}{y}\) |
| Multiply the fractions. | \(=\dfrac{xxx}{yyy}\) |
| Write with exponents. | \(=\dfrac{x^3}{y^3}\) |
Notice that the exponent applies to both the numerator and the denominator.
We see that \(\left(\dfrac{x}{y}\right)^3\) is \(\dfrac{x^3}{y^3}\).
This leads to the Quotient to a Power Property for Integer Exponents . We can say that the exponent distributes over division as well as multiplication.
If \(a\) and \(b\) are real numbers, \(b\neq 0\), and \(m\) is an integer, then
\[\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}. \nonumber \]
To raise a fraction to a power, raise the numerator and denominator to that power.
Simplify each expression:
a. \(\left(\dfrac{b}{3}\right)^4\)
b. \(\left(\dfrac{k}{j}\right)^{−3}\)
c. \(\left(\dfrac{2xy^2}{z}\right)^3\)
d. \(\left(\dfrac{4p^{−3}}{q^2}\right)^2\)
Solution
a.
| \(\quad\left(\dfrac{b}{3}\right)^4\) | |
|---|---|
| Use the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}\). | \(=\dfrac{b^4}{3^4}\) |
| Simplify. | \(=\dfrac{b^4}{81}\) |
b.
| \(\quad\left(\dfrac{k}{j}\right)^{−3}\) | |
|---|---|
| Raise the numerator and denominator to the power. | \(=\dfrac{k^{-3}}{j^{-3}}\) |
| Use the definition of negative exponent, \(a^{−n}=\dfrac{1}{a^n}\) and \(\dfrac{1}{a^{-n}}=a^n\). | \(=\dfrac{1}{k^3}\cdot\dfrac{j^3}{1}\) |
| Multiply. | \(=\dfrac{j^3}{k^3}\) |
c.
| \(\quad\left(\dfrac{2xy^2}{z}\right)^3\) | |
|---|---|
| Use the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}\). | \(=\dfrac{(2xy^2)^3}{z^3}\) |
| Use the Product to a Power Property, \((ab)^m=a^mb^m\), and the Power Property, \((a^m)^n=a^{mn}\). | \(=\dfrac{8x^3y^6}{z^3}\) |
d.
| \(\quad\left(\dfrac{4p^{−3}}{q^2}\right)^2\) | |
|---|---|
| Use the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}\). | \(=\dfrac{(4p^{−3})^2}{(q^2)^2}\) |
| Use the Product to a Power Property, \((ab)^m=a^mb^m\). | \(=\dfrac{16p^{−6}}{q^4}\) |
| Use the definition of negative exponent, \(a^{−n}=\dfrac{1}{a^n}\). | \(=\dfrac{16}{q^4}\cdot\dfrac{1}{p^6}\) |
| Simplify. | \(=\dfrac{16}{p^6q^4}\) |
Simplify each expression:
a. \(\left(\dfrac{p}{10}\right)^4\)
b. \(\left(\dfrac{m}{n}\right)^{−7}\)
c. \(\left(\dfrac{3ab^3}{c^2}\right)^4\)
d. \(\left(\dfrac{3x^{−2}}{y^3}\right)^3\)
- Answer
-
a. \(\dfrac{p^4}{10000}\)
b. \(\dfrac{n^7}{m^7}\)
c. \(\dfrac{81a^4b^{12}}{c^8}\)
d. \(\dfrac{27}{x^6y^9}\)
Simplify each expression:
a. \(\left(\dfrac{−2}{q}\right)^3\)
b. \(\left(\dfrac{w}{x}\right)^{−4}\)
c. \(\left(\dfrac{xy^3}{3z^2}\right)^2\)
d. \(\left(\dfrac{2m^{−2}}{n^{−2}}\right)^3\)
- Answer
-
a. \(\dfrac{−8}{q^3}\)
b. \(\dfrac{x^4}{w^4}\)
c. \(\dfrac{x^2y^6}{9z^4}\)
d. \( \dfrac{8n^6}{m^6}\)
We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties. We note that there are many ways to simplify the expressions, but the final simplification should be equivalent.
If \(a\) and \(b\) are real numbers, and \(m\) and \(n\) are integers, then
| Definition | Description |
|---|---|
| Definition of Zero Exponent | \(a^0=1,a \neq 0\) |
| Definition of Negative Exponents | \(a^{−n}=\dfrac{1}{a^n}\), or equivalently, \(\dfrac{1}{a^{−n}}=a^n\) |
| Property | Description |
|---|---|
| Product Property | \(a^m·a^n=a^{m+n}\) |
| Power Property | \((a^m)^n=a^{m·n}\) |
| Product to a Power | \((ab)^n=a^nb^n\) |
| Quotient Property | \(\dfrac{a^m}{a^n}=a^{m−n},a\neq 0\) |
| Quotient to a Power Property | \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m},b \neq 0 \) |
| Quotient to a Negative Exponent | \(\left(\dfrac{a}{b}\right)^{−n}=\left(\dfrac{b}{a}\right)^n\) |
Simplify each expression by applying several properties:
a. \((3x^2y)^4(2xy^2)^3\)
b. \(\dfrac{(x^3)^4(x^{−2})^5}{(x^6)^5}\)
c. \(\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2 \left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}\)
Solution
a.
| \(\quad (3x^2y)^4(2xy^2)^3\) | |
|---|---|
| Use the Product to a Power Property, \((ab)^m=a^mb^m\). | \(=(3^4x^8y^4)(2^3x^3y^6)\) |
| Simplify. | \(=(81x^8y^4)(8x^3y^6)\) |
| Use the Commutative Property. | \(=81\cdot 8x^8x^3y^4y^6\) |
| Multiply the constants and add the exponents. | \(=648x^{11}y^{10}\) |
b.
| \(\quad\dfrac{(x^3)^4(x^{−2})^5}{(x^6)^5}\) | |
|---|---|
| Use the Power Property, \((a^m)^n=a^{m·n}\). | \(=\dfrac{x^{12}x^{−10}}{x^{30}}\) |
| Add the exponents in the numerator. | \(=\dfrac{x^2}{x^{30}}\) |
| Use the Quotient Property, \(\dfrac{a^m}{a^n}=\dfrac{1}{a^{n−m}}\) | \(=\dfrac{1}{x^{28}}\) |
c.
| \(\quad\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2 \left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}\) | |
|---|---|
| Simplify inside the parentheses first. | \(=\left(\dfrac{2y^4}{x^2}\right)^2\left(\dfrac{12y^4}{x^2}\right)^{−1}\) |
| Use the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}\). | \(=\dfrac{(2y^4)^2}{(x^2)^2}\dfrac{(12y^4)^{−1}}{(x^2)^{−1}}\) |
| Use the Product to a Power Property, \((ab)^m=a^mb^m\). | \(=\dfrac{4y^8}{x^4}·\dfrac{12^{−1}y^{−4}}{x^{−2}}\) |
| Simplify. | \(=\dfrac{4y^4}{12x^2}\) |
| Simplify. | \(=\dfrac{y^4}{3x^2}\) |
Simplify each expression:
a. \((c^4d^2)^5(3cd^5)^4\)
b. \(\dfrac{(a^{−2})^3(a^2)^4}{(a^4)^5}\)
c. \(\left(\dfrac{3xy^2}{x^2y^{−3}}\right)^2\)
- Answer
-
a. \(81c^{24}d^{30}\)
b. \(\dfrac{1}{a^{18}}\)
c. \(\dfrac{9y^{10}}{x^2}\)
Simplify each expression:
a. \((a^3b^2)^6(4ab^3)^4\)
b. \(\dfrac{(p^{−3})^4(p^5)^3}{(p^7)^6}\)
c. \(\left(\dfrac{4x^3y^2}{x^2y^{−1}}\right)^2\left(\dfrac{8xy^{−3}}{x^2y}\right)^{−1}\)
- Answer
-
a. \(256a^{22}b^{24}\)
b. \(\dfrac{1}{p^{39}}\)
c. \(2x^3y^{10}\)
- Give an example of distributing division over subtraction.
- Give an example to show that you can not distribute division over multiplication.
- How is the negative exponent related to reciprocals? Give an example.
- How are positive and negative exponents used in science to express large or small numbers?
- What is the purpose in writing numbers this way?
a. Simplify \(\left(\dfrac{3a^{−3}}{b^{−5}}\right)^{−3}\). Write your final answer with positive exponents only.
b. Simplify \(\dfrac{36x^5y^{10}}{70x^{15}y^5}\). Write your final answer with positive exponents only.
Key Concepts
-
Exponential Notation
This is read \(a\) to the \(m^{\mathrm{th}}\) power.
In the expression \(a^m\), the exponent \(m\) (when positive) tells us how many times we use the base \(a\) as a factor. - Zero Exponent (Definition) If \(a\) is a non-zero number, then \(a^0=1\).
- Negative Exponent (Definition) If \(n\) is an integer and \(a\neq 0\), then \(a^{−n}=\dfrac{1}{a^n}\) or, equivalently, \(\dfrac{1}{a^{−n}}=a^n\).
-
Product Property for Exponents
If \(a\) is a real number and \(m\) and \(n\) are integers, then\[a^ma^n=a^{m+n} \nonumber \]
To multiply with like bases, add the exponents. -
Quotient Property for Exponents
If \(a\) is a real number, \(a\neq 0\), and \(m\) and \(n\) are integers, then\[\begin{array} {lllll} {\dfrac{a^m}{a^n}=a^{m−n},} &{m>n} &{\text{and}} &{\dfrac{a^m}{a^n}=\dfrac{1}{a^{n−m}},} &{n>m}\\ \nonumber \end{array}\]
-
Quotient to a Negative Exponent Property
If \(a\) and \(b\) are real numbers, \(a\neq 0\), \(b\neq 0\) and \(n\) is an integer, then\[\left(\dfrac{a}{b}\right)^{−n}=\left(\dfrac{b}{a}\right)^n\nonumber \]
-
Power Property for Exponents
If \(a\) is a real number and \(m\) and \(n\) are integers, then\[(a^m)^n=a^{mn}\nonumber \]
To raise a power to a power, multiply the exponents. -
Product to a Power Property for Exponents
If \(a\) and \(b\) are real numbers and \(m\) is a whole number, then\[(ab)^m=a^mb^m \nonumber \]
To raise a product to a power, raise each factor to that power. -
Quotient to a Power Property for Exponents
If \(a\) and \(b\) are real numbers, \(b\neq 0\), and \(m\) is an integer, then\[\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m} \nonumber \]
To raise a fraction to a power, raise the numerator and denominator to that power. -
Summary of Exponent Properties
If \(a\) and \(b\) are real numbers, and \(m\) and \(n\) are integers, then -
Property Description Definition of Zero Exponent \(a^0=1,a \neq 0\) Definition of Negative Exponents \(a^{−n}=\dfrac{1}{a^n}\), or equivalently, \(\dfrac{1}{a^{−n}}=a^n\) -
Property Description Product Property \(a^m·a^n=a^{m+n}\) Power Property \((a^m)^n=a^{m·n}\) Product to a Power \((ab)^n=a^nb^n\) Quotient Property \(\dfrac{a^m}{a^n}=a^{m−n},a\neq 0\) Quotient to a Power Property \(\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m},b \neq 0 \) Quotient to a Negative Exponent \(\left(\dfrac{a}{b}\right)^{−n}=\left(\dfrac{b}{a}\right)^n\)


4.6: Make use of small blocks of time
Through using smaller blocks of time you can cover material in chunks (more on the next page) and not have to worry about the larger whole. A mistake that many people make is that they try to cram information into their minds in one large session. This isn’t a successful strategy for most students.
Look for smaller blocks of time to study. If you are a public transit user, you can likely spend 20 minutes on your bus ride to read or review for your upcoming class or exam. You could even listen to an audio recording of your notes. In the evening, instead of watching three episodes of your favourite TV show, you could watch one and spend the remaining time preparing for your studies. Going out to eat often? Consider making something simple at home that you could put in the oven to cook without needing tending to; that time could be used doing some work for class and still leave you time for other activities once dinner is done.
Making time for your studies can be overwhelming. The following video introduces you to ways to use smaller blocks of time to get your tasks done, while not using up numerous hours at once.
Click on the video to learn more as you continue down this pathway. Answer the questions (by clicking on the icons that appear) to further your learning. When you are finished, go to the next chapter to continue.
An interactive or media element has been excluded from this version of the text. You can view it online here:
https://kpu.pressbooks.pub/learningtolearnonline/?p=84
Video Transcript: Using Small Blocks of Time Productively
One trick to balancing work and study is taking advantage of small blocks of time to get things done.
In this video, you will consider the small blocks of time in your schedule, and identify strategies to increase your productivity during these moments in your day.
Often, we think we need to have a lot of time available for study, or we think that we can only study at home or in the library. By adjusting your thinking, you’ll be able to open up additional productive learning time.
- Do you commute by transit? Though it wouldn’t be ideal to try to master detailed or complicated reading material on the bus, perhaps you can do some initial scanning or skimming while in transit, to prepare yourself for class or deeper reading later.
- Consider creating flash cards for material that you need to learn. You can take a set of flash cards with you and work whenever a few minutes become available. If you use one of the many flash card or self-testing apps available on your phone, you’ll be able to easily pull out your phone and make use of those small blocks of time.
- Self-testing is one of the most effective ways to learn. Create a list of study questions for your course. Pull out the list when you have time available, and review a few questions. Keep track of those you answer correctly, and those you need to study more.
- Does your course include access to online videos that explain and review key concepts? Watch a video or two to review, or to improve your understanding of a key course idea.
- Some courses also include access to online self-study questions. Try answering a few review questions in your spare moments. These online quizzes usually provide immediate feedback on what you understand, and what you should study further.
- Do you like to learn by listening? Make an audio recording of the important points you want to remember, and listen while you commute or exercise. Maybe audio books are for you – are any of your course materials available in this format?


The Spore Stain
Name: _____________________________________________________
Course Section: ______________________________________________
The following activity will teach students how to complete an endospore stain and differentiate between vegetative cells and endospores. Read the following procedure carefully before starting the activity.
|
Materials
|
-
2 microscope slides
-
Wax pencil
-
Bacillus cereus
(24-hour) and
Bacillus cereus
(5-day)
-
Malachite green stain
-
Safranin stain
-
DI water
-
Bibulous paper
-
Staining tray and slide stand
-
Inoculation loop
-
Hot plate
-
Clothespin
-
Forceps
-
1000 ml beaker
-
Timer
-
Incinerator
-
Microscopes: lens paper, lens cleaner, immersion oil
|
|
Procedure
|
|
Step 1: Prepare A Bacterial Smear
-
Get into groups of 2.
-
Label the edge of the microscope slide with 1 person's initials and the bacterial species. You may abbreviate the name of the bacteria so it fits on the slide. Try to write small.
-
On the bottom of the slide, you may draw a circle in the middle of the slide with sharpie a or wax pencil so you know where to make your smear. This is optional.
-
Sterilize your inoculation loop and use it to pick up a small amount of DI water. Place the water on top of the slide, in the middle or, in the circle if you drew one.
-
Sterilize your loop again and pick up a small bacterial colony from the stock plate.
-
Gently, mix the bacteria into the water drop on the slide. You want to spread the water and bacteria out to about the size of a coin.
-
Let the water and bacteria smear completely air dry. The slide should look dry and "crusty" before moving to Step 2.
|
|
Step 2: Heat Fixation
-
Once the bacterial smear is completely dry, use a clothespin to clip the end of your slide.
-
Then, place your slide directly on a hot plate for 30 seconds.
-
After 30 seconds, place your slide onto the slide stand in your staining tray. Don't touch the slide, it will be hot. Don't place the slide on the cold table, the glass slide could crack.
-
Once your slide is completely cool, move on to Step 3.
|
|
Step 3: Spore Staining
-
Primary Staining
-
Take your heat-fixed slide to the fume hood.
-
Once the water is boiling, place your slide on the slide rack above the boiling water.
-
Cover the smear area of your slide with a square piece of bibulous paper.
-
Carefully apply the malachite green stain to the bibulous paper.
-
Steam with the stain on the slide for 5 minutes while continuously applying more stain so the bibulous paper never dries out.
-
After 5 minutes remove the paper with forceps and discard it in the trash bin.
-
Using a clothespin take your slide back to your lab station and rinse with water.
-
Decolorizer, Wash Step
-
Hold the slide at a 45° angle and wash the slide with a gentle DI water rinse.
-
Counterstain
-
Add a few drops of safranin to the smear.
-
Leave the stain for 1 minute.
-
Hold the slide at a 45° angle and rinse with DI water.
-
Place the slide between 2 sheets of bibulous paper and gently pat dry. Do not wipe the slide to dry it.
-
Microscopy
-
When the slide is dry, observe your specimen using oil immersion.
-
Repeat Steps 1 through 3 for the remaining bacterial species.
-
When both slides are done, clean your microscope and place it back in the cabinet.
-
Rinse your staining tray in a sink and put all materials back in their labeled locations.
|
Results
|
Species
|
Vegetative Cells (+/-)
|
Endospores (+/-)
|
Spore Location
|
Bacterial Morphology and Arrangement
|
|
Bacillus cereus
(24-hour)
|
|
|
|
|
|
Bacillus cereus
(5-day)
|
|
|
|
|
General Questions
1. Fill in the blanks: ___________________ are the metabolically active cells that stain ___________________ during the spore stain procedure. ___________________ are mainly metabolically inactive and stain ___________________.
2. Why might a species of bacteria form endospores? What is the advantage of forming spores for bacteria?
3. What genera of bacteria are known to have pathogenic (i.e., disease-causing) species and are spore-forming genera?
4. What is the purpose of the steam in the spore stain?
5. Which step serves as the differential step? Explain your answer.
6. What role do broad-spectrum antibiotics play in the development of
C. difficile
infections, and how do they affect the normal gut flora?
7. Why is the ability of the endospore stain to identify
C. difficile
endospores important for infection control in healthcare settings?


38.3: Joints and Skeletal Movement
Skills to Develop
- Classify the different types of joints on the basis of structure
- Explain the role of joints in skeletal movement
The point at which two or more bones meet is called a joint , or articulation . Joints are responsible for movement, such as the movement of limbs, and stability, such as the stability found in the bones of the skull.
Classification of Joints on the Basis of Structure
There are two ways to classify joints: on the basis of their structure or on the basis of their function. The structural classification divides joints into bony, fibrous, cartilaginous, and synovial joints depending on the material composing the joint and the presence or absence of a cavity in the joint.
Fibrous Joints
The bones of fibrous joints are held together by fibrous connective tissue. There is no cavity, or space, present between the bones and so most fibrous joints do not move at all, or are only capable of minor movements. There are three types of fibrous joints: sutures, syndesmoses, and gomphoses. Sutures are found only in the skull and possess short fibers of connective tissue that hold the skull bones tightly in place (Figure \(\PageIndex{1}\)).
Syndesmoses are joints in which the bones are connected by a band of connective tissue, allowing for more movement than in a suture. An example of a syndesmosis is the joint of the tibia and fibula in the ankle. The amount of movement in these types of joints is determined by the length of the connective tissue fibers.  The tooth is connected to the socket by a connective tissue referred to as the periodontal ligament.
Cartilaginous Joints
Cartilaginous joints are joints in which the bones are connected by cartilage. There are two types of cartilaginous joints: synchondroses and symphyses. In a synchondrosis , the bones are joined by hyaline cartilage. Synchondroses are found in the epiphyseal plates of growing bones in children. In symphyses , hyaline cartilage covers the end of the bone but the connection between bones occurs through fibrocartilage. Symphyses are found at the joints between vertebrae. Either type of cartilaginous joint allows for very little movement.
Synovial Joints
Synovial joints are the only joints that have a space between the adjoining bones (Figure \(\PageIndex{3}\)). This space is referred to as the synovial (or joint) cavity and is filled with synovial fluid. Synovial fluid lubricates the joint, reducing friction between the bones and allowing for greater movement. The ends of the bones are covered with articular cartilage, a hyaline cartilage, and the entire joint is surrounded by an articular capsule composed of connective tissue that allows movement of the joint while resisting dislocation. Articular capsules may also possess ligaments that hold the bones together. Synovial joints are capable of the greatest movement of the three structural joint types; however, the more mobile a joint, the weaker the joint. Knees, elbows, and shoulders are examples of synovial joints.
Classification of Joints on the Basis of Function
The functional classification divides joints into three categories: synarthroses, amphiarthroses, and diarthroses. A synarthrosis is a joint that is immovable. This includes sutures, gomphoses, and synchondroses. Amphiarthroses are joints that allow slight movement, including syndesmoses and symphyses. Diarthroses are joints that allow for free movement of the joint, as in synovial joints.
Movement at Synovial Joints
The wide range of movement allowed by synovial joints produces different types of movements. The movement of synovial joints can be classified as one of four different types: gliding, angular, rotational, or special movement.
Gliding Movement
Gliding movements occur as relatively flat bone surfaces move past each other. Gliding movements produce very little rotation or angular movement of the bones. The joints of the carpal and tarsal bones are examples of joints that produce gliding movements.
Angular Movement
Angular movements are produced when the angle between the bones of a joint changes. There are several different types of angular movements, including flexion, extension, hyperextension, abduction, adduction, and circumduction. Flexion , or bending, occurs when the angle between the bones decreases. Moving the forearm upward at the elbow or moving the wrist to move the hand toward the forearm are examples of flexion. Extension is the opposite of flexion in that the angle between the bones of a joint increases. Straightening a limb after flexion is an example of extension. Extension past the regular anatomical position is referred to as hyperextension . This includes moving the neck back to look upward, or bending the wrist so that the hand moves away from the forearm.
Abduction occurs when a bone moves away from the midline of the body. Examples of abduction are moving the arms or legs laterally to lift them straight out to the side. Adduction is the movement of a bone toward the midline of the body. Movement of the limbs inward after abduction is an example of adduction. Circumduction is the movement of a limb in a circular motion, as in moving the arm in a circular motion.
Rotational Movement
Rotational movement is the movement of a bone as it rotates around its longitudinal axis. Rotation can be toward the midline of the body, which is referred to as medial rotation , or away from the midline of the body, which is referred to as lateral rotation . Movement of the head from side to side is an example of rotation.
Special Movements
Some movements that cannot be classified as gliding, angular, or rotational are called special movements. Inversion involves the soles of the feet moving inward, toward the midline of the body. Eversion is the opposite of inversion, movement of the sole of the foot outward, away from the midline of the body. Protraction is the anterior movement of a bone in the horizontal plane. Retraction occurs as a joint moves back into position after protraction. Protraction and retraction can be seen in the movement of the mandible as the jaw is thrust outwards and then back inwards. Elevation is the movement of a bone upward, such as when the shoulders are shrugged, lifting the scapulae. Depression is the opposite of elevation—movement downward of a bone, such as after the shoulders are shrugged and the scapulae return to their normal position from an elevated position. Dorsiflexion is a bending at the ankle such that the toes are lifted toward the knee. Plantar flexion is a bending at the ankle when the heel is lifted, such as when standing on the toes. Supination is the movement of the radius and ulna bones of the forearm so that the palm faces forward. Pronation is the opposite movement, in which the palm faces backward. Opposition is the movement of the thumb toward the fingers of the same hand, making it possible to grasp and hold objects.
Types of Synovial Joints
Synovial joints are further classified into six different categories on the basis of the shape and structure of the joint. The shape of the joint affects the type of movement permitted by the joint (Figure \(\PageIndex{4}\)). These joints can be described as planar, hinge, pivot, condyloid, saddle, or ball-and-socket joints.
Planar Joints
Planar joints have bones with articulating surfaces that are flat or slightly curved faces. These joints allow for gliding movements, and so the joints are sometimes referred to as gliding joints. The range of motion is limited in these joints and does not involve rotation. Planar joints are found in the carpal bones in the hand and the tarsal bones of the foot, as well as between vertebrae (Figure \(\PageIndex{5}\)).
Hinge Joints
In hinge joints , the slightly rounded end of one bone fits into the slightly hollow end of the other bone. In this way, one bone moves while the other remains stationary, like the hinge of a door. The elbow is an example of a hinge joint. The knee is sometimes classified as a modified hinge joint (Figure \(\PageIndex{6}\)).
Pivot Joints
Pivot joints consist of the rounded end of one bone fitting into a ring formed by the other bone. This structure allows rotational movement, as the rounded bone moves around its own axis. An example of a pivot joint is the joint of the first and second vertebrae of the neck that allows the head to move back and forth (Figure \(\PageIndex{7}\)). The joint of the wrist that allows the palm of the hand to be turned up and down is also a pivot joint.
Condyloid Joints
Condyloid joints consist of an oval-shaped end of one bone fitting into a similarly oval-shaped hollow of another bone (Figure \(\PageIndex{8}\)). This is also sometimes called an ellipsoidal joint. This type of joint allows angular movement along two axes, as seen in the joints of the wrist and fingers, which can move both side to side and up and down.
Saddle Joints
Saddle joints are so named because the ends of each bone resemble a saddle, with concave and convex portions that fit together. Saddle joints allow angular movements similar to condyloid joints but with a greater range of motion. An example of a saddle joint is the thumb joint, which can move back and forth and up and down, but more freely than the wrist or fingers (Figure \(\PageIndex{9}\)).
Ball-and-Socket Joints
Ball-and-socket joints possess a rounded, ball-like end of one bone fitting into a cuplike socket of another bone. This organization allows the greatest range of motion, as all movement types are possible in all directions. Examples of ball-and-socket joints are the shoulder and hip joints (Figure \(\PageIndex{10}\)).
Link to Learning
Watch this animation showing the six types of synovial joints.
Career Connection: Rheumatologist
Rheumatologists are medical doctors who specialize in the diagnosis and treatment of disorders of the joints, muscles, and bones. They diagnose and treat diseases such as arthritis, musculoskeletal disorders, osteoporosis, and autoimmune diseases such as ankylosing spondylitis and rheumatoid arthritis.
Rheumatoid arthritis (RA) is an inflammatory disorder that primarily affects the synovial joints of the hands, feet, and cervical spine. Affected joints become swollen, stiff, and painful. Although it is known that RA is an autoimmune disease in which the body’s immune system mistakenly attacks healthy tissue, the cause of RA remains unknown. Immune cells from the blood enter joints and the synovium causing cartilage breakdown, swelling, and inflammation of the joint lining. Breakdown of cartilage causes bones to rub against each other causing pain. RA is more common in women than men and the age of onset is usually 40–50 years of age.
Rheumatologists can diagnose RA on the basis of symptoms such as joint inflammation and pain, X-ray and MRI imaging, and blood tests. Arthrography is a type of medical imaging of joints that uses a contrast agent, such as a dye, that is opaque to X-rays. This allows the soft tissue structures of joints—such as cartilage, tendons, and ligaments—to be visualized. An arthrogram differs from a regular X-ray by showing the surface of soft tissues lining the joint in addition to joint bones. An arthrogram allows early degenerative changes in joint cartilage to be detected before bones become affected.
There is currently no cure for RA; however, rheumatologists have a number of treatment options available. Early stages can be treated with rest of the affected joints by using a cane or by using joint splints that minimize inflammation. When inflammation has decreased, exercise can be used to strengthen the muscles that surround the joint and to maintain joint flexibility. If joint damage is more extensive, medications can be used to relieve pain and decrease inflammation. Anti-inflammatory drugs such as aspirin, topical pain relievers, and corticosteroid injections may be used. Surgery may be required in cases in which joint damage is severe.
Summary
The structural classification of joints divides them into bony, fibrous, cartilaginous, and synovial joints. The bones of fibrous joints are held together by fibrous connective tissue; the three types of fibrous joints are sutures, syndesomes, and gomphoses. Cartilaginous joints are joints in which the bones are connected by cartilage; the two types of cartilaginous joints are synchondroses and symphyses. Synovial joints are joints that have a space between the adjoining bones. The functional classification divides joints into three categories: synarthroses, amphiarthroses, and diarthroses. The movement of synovial joints can be classified as one of four different types: gliding, angular, rotational, or special movement. Gliding movements occur as relatively flat bone surfaces move past each other. Angular movements are produced when the angle between the bones of a joint changes. Rotational movement is the movement of a bone as it rotates around its own longitudinal axis. Special movements include inversion, eversion, protraction, retraction, elevation, depression, dorsiflexion, plantar flexion, supination, pronation, and opposition. Synovial joints are also classified into six different categories on the basis of the shape and structure of the joint: planar, hinge, pivot, condyloid, saddle, and ball-and-socket.
Glossary
- abduction
- when a bone moves away from the midline of the body
- adduction
- movement of the limbs inward after abduction
- amphiarthrosis
- joint that allows slight movement; includes syndesmoses and symphyses
- angular movement
- produced when the angle between the bones of a joint changes
- ball-and-socket joint
- joint with a rounded, ball-like end of one bone fitting into a cuplike socket of another bone
- cartilaginous joint
- joint in which the bones are connected by cartilage
- circumduction
- movement of a limb in a circular motion.
- condyloid joint
- oval-shaped end of one bone fitting into a similarly oval-shaped hollow of another bone
- depression
- movement downward of a bone, such as after the shoulders are shrugged and the scapulae return to their normal position from an elevated position; opposite of elevation
- diarthrosis
- joint that allows for free movement of the joint; found in synovial joints
- dorsiflexion
- bending at the ankle such that the toes are lifted toward the knee
- elevation
- movement of a bone upward, such as when the shoulders are shrugged, lifting the scapulae
- eversion
- movement of the sole of the foot outward, away from the midline of the body; opposite of inversion
- extension
- movement in which the angle between the bones of a joint increases; opposite of flexion
- fibrous joint
- joint held together by fibrous connective tissue
- flexion
- movement in which the angle between the bones decreases; opposite of extension
- gliding movement
- when relatively flat bone surfaces move past each other
- gomphosis
- the joint in which the tooth fits into the socket like a peg
- hinge joint
- slightly rounded end of one bone fits into the slightly hollow end of the other bone
- hyperextension
- extension past the regular anatomical position
- inversion
- soles of the feet moving inward, toward the midline of the body
- joint
- point at which two or more bones meet
- lateral rotation
- rotation away from the midline of the body
- medial rotation
- rotation toward the midline of the body
- opposition
- movement of the thumb toward the fingers of the same hand, making it possible to grasp and hold objects
- plantar flexion
- bending at the ankle such that the heel is lifted, such as when standing on the toes
- planar joint
- joint with bones whose articulating surfaces are flat
- pivot joint
- joint with the rounded end of one bone fitting into a ring formed by the other bone
- pronation
- movement in which the palm faces backward
- protraction
- anterior movement of a bone in the horizontal plane
- retraction
- movement in which a joint moves back into position after protraction
- rotational movement
- movement of a bone as it rotates around its own longitudinal axis
- saddle joint
- joint with concave and convex portions that fit together; named because the ends of each bone resemble a saddle
- supination
- movement of the radius and ulna bones of the forearm so that the palm faces forward
- suture
- short fiber of connective tissue that holds the skull bones tightly in place; found only in the skull
- synarthrosis
- joint that is immovable
- symphysis
- hyaline cartilage covers the end of the bone, but the connection between bones occurs through fibrocartilage; symphyses are found at the joints between vertebrae
- synchondrosis
- bones joined by hyaline cartilage; synchondroses are found in the epiphyseal plates of growing bones in children
- syndesmosis
- joint in which the bones are connected by a band of connective tissue, allowing for more movement than in a suture
- synovial joint
- only joint that has a space between the adjoining bones


17.1: Introduction
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Learning Outcomes
After reading this chapter, you should be able to answer these questions:
- How can you enhance your interpersonal skills?
- Why is learning to plan so important in school and in the real world?
- What skills should you develop in school that can transfer easily to your professional life and make it a success?
- What are some strategies that will help you find, keep, and advance in your dream job?
- What key attributes do employers look for when interviewing job candidates?

Never Quit Until You Have Your Degree in Hand!
What makes someone a winner in life? A winner is someone who goes through the various stages of life satisfied in knowing that they have done their best: their best at work, home, and in all pursuits of life. A big part of having a happy life is pursuing a career that offers job satisfaction and financial rewards. If you are going to “be all that you can be,” you need a good education.
A college degree unlocks doors to economic opportunity.
Why get a degree?
- Get and keep a better job. Because the world is changing rapidly and many jobs rely on new technology, more jobs require education beyond high school. With a college education, you will have more jobs from which to choose.
- Earn more money. People who go to college usually earn more than those who do not. Currently, a bachelor’s degree is worth a minimum of $20,000 a year more than a high school diploma. If your career spans 45 years, you could earn close to $1 million more than a high school graduate.
- Get a good start in life. A business college education helps you acquire a wide range of knowledge in many subjects as well as an advanced understanding of your specialized area of business. College also trains you to express your thoughts clearly in speech and in writing and to make informed decisions.
Simply stated, a degree in business gives you the chance to achieve the quality of life you deserve. The lifestyle, the new friends, the purchasing power of a degree won’t guarantee happiness but will put you well on the road to finding it.