4.6: Intersections of Inequality
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Explain and give examples of intersectionality.
- Discuss how accumulated wealth creates systems of social inequality.
- Give examples of the ways that governing bodies can negatively impact the lived experiences of individuals.
- Explain caste systems as a type of intersection of political, economic, and racial inequalities.
- Explain implicit mentalities around poverty, wealth, and equity disparities.
Intersectionality
When thinking about social inequalities, it is useful to conceptualize race alongside other characteristics. Intersectionality is the observation that one’s class, race, sexuality, age, and ability can all define and complicate experiences. The concept of intersectionality can be traced back to pre–Civil War America, when Sojourner Truth made her “Ain’t I a Woman” speech in 1851 at the Ohio Women’s Convention in Akron, Ohio, addressing the exclusion of Black women from the fight for women’s rights. However, the term intersectionality was officially coined by critical race theorist and legal scholar Kimberlé Crenshaw (1989) in the context of discussing Black feminism. Crenshaw argued that the experience of being a Black woman could not be understood in independent terms of either being Black or being a woman; instead, it needed to include interactions between the identities, which often reinforce one another. Intersectionality discredits the notion that one single aspect of identity—race, for example—can capture the multidimensional nature of people’s experiences of oppression. In other words, intersectionality emphasizes the ways in which identities pertaining to features such as race, gender, and class interact to impact people’s lives.
Anthropologist Faye Harrison, coeditor of African-American Pioneers in Anthropology (1999), has done extensive work on intersectionality. She argues that “race is always lived in class- and gender-specific ways” (Harrison 1995, 63). For instance, the lived experience of a woman of color will be different from that of a White woman. Even though they both experience oppression from patriarchal systems, a woman of color has the added intersection of race, impacted by her identity as a woman.
Much of the work on intersectionality has come out of a critique of the original feminist movement, which sometimes generalized women’s experiences as monolithic (Hill Collins 2000; A. Y. Davis 1981; McCall 2005; Sacks 1989). Feminist and women’s studies scholar Chandra Mohanty (1984) criticized the White-middle-class-based approach of previous feminist authors, arguing not only that women of color don’t need White women to save them but that their experiences are vastly different. By incorporating race with gender and class, feminist scholars have illustrated how experiences of race are dynamic.
In the collection of studies of race, class, and gender that occurred around the turn of the 21st century, anthropologist Leith Mullings (2002) developed the concept of the Sojourner syndrome to capture the interlocking ways in which race, class, gender, and resistance to oppression shape Black women’s bodies and biology. The Sojourner syndrome emphasizes that race, class, and gender are not necessarily multiplied to mean more oppression, but they change the ways people experience oppression. In the Harlem Birthright Project, funded by the Centers for Disease Control and Prevention (CDC) to study racial disparities in health, Mullings uses the Sojourner syndrome to argue that Black women, because of intersecting structural inequalities, are forced to do more work than either their White female or Black male counterparts, which increases their stress levels and negatively impacts their health.
Another way intersectional identities can compound oppression is captured by the term misogynoir . Misogyny is the socialized prejudice against women and feminine characteristics.  Misogynoir is the intersection of the systems of sexism and racism experienced by Black women. More recently, Bailey has written about Black women’s digital resistance to misogynoir on YouTube, Facebook, and other online platforms (2021).
In addition to creating challenges to the status quo, intersectionality can also inspire creative opportunities for new perspectives and new role models. On January 20, 2021, former senator Kamala Harris was inaugurated as the 49th vice president of the United States. Not only is she the first female vice president and the highest-ranking female official in US history, but her ethnic and racial background makes her the first Black American and the first Asian American person to hold this office. When she broke these “glass ceilings” (barriers to promotion that often affect women and members of minority groups), she was celebrated as a role model for many. There is even an unofficial Twitter fan group that calls itself “ The #Khive Movement ” as well as other pro-Harris groups inspired by her example (e.g., Mamas for Momala). Her supporters frequently cite her intersectionality as an inspiring triumph that allows for new voices representing diverse groups in our society.
Overall, the Biden administration has pledged to have “the most diverse cabinet in American history” (see the “ Biden Diversity Tracker ”). On October 28, 2021, President Joe Biden appointed Sara Minkara as the US special advisor on international disability rights. In this foreign policy role, Minkara, who lost her eyesight at the age of seven, will promote and protect the rights of people with disabilities, again representing diverse voices of historically underrepresented groups.
Profiles in Anthropology
Dr. Yolanda T. Moses (1946–)
Personal History: Yolanda Moses was born in Washington, DC, but spent most of her childhood in Southern California. An active participant in the civil rights movement in the 1960s, she was inspired to pursue a doctoral program in anthropology after meeting Margaret Mead.
Area of Anthropology: Dr. Moses is currently professor of anthropology and associate vice chancellor for diversity, excellence, and equity at the University of California, Riverside. Her research focuses on the origins of social inequalities, relying on both comparative ethnographic and survey methods. She has examined gender and class disparities in the Caribbean, East Africa, and the United States. Dr. Moses’s most recent research has focused on issues of diversity and change in universities and colleges in the United States, India, Europe, and South Africa.
Accomplishments in the Field: Dr. Moses has served as president of the American Anthropological Association (1995–1997), the City University of New York’s City College of New York (1993–1999), and the American Association for Higher Education (2000–2003). She received the Donna Shavlik Award for leadership and mentoring of women from the American Council on Education in 2007 and the Franz Boas Award for Exemplary Service to Anthropology from the American Anthropological Association in 2015.
Importance of Her Work: Dr. Moses has received numerous grants from the Ford Foundation, the National Science Foundation, and the National Endowment of the Humanities. These grants have been awarded for projects examining the experiences of faculty who are women of color, questions of leadership and diversity in higher education, and, more broadly, race and human variation. She is a coauthor of Race: Are We So Different? and was influential in the RACE Project, a national public education project on race and human variation sponsored by the American Anthropological Association.
Global Inequalities
Anthropologists, along with other social scientists, recognize that all social systems and structures have developed through a multitude of decisions made by people with social, political, and economic power as well as through the daily interactions and imaginations of individuals. The current world system is the result of an amalgamation of events and historical forces that led humanity, step by step, to the world as it is today. Social systems and social structures are constructed and governed by the people who live within them; they are not ahistorical, and they are not unchanging. Capitalism is an economic system, but it is also the result of the ways in which people and groups interact with each other and with the natural world. Presidents elected by slim margins, compromises that benefited one political party over the other, and responses to natural disasters and other events, some of which may have seemed inconsequential at the time, all played a role in creating the current reality. Structures exist and order the world, but they do not exist outside of it.
When talking about the effect of capitalism, it is important to recognize the ways in which these systems of inequality can intersect to both benefit the powerful and exploit the poor. Wealth inequalities and capital accumulation have deeply impacted and continued to impact cultures around the world, leaving almost none untouched. There are two broad forces that shape this movement of economic capital. One of those forces, which encourages further and further accumulation of wealth within a single family, is intergenerational wealth . Intergenerational wealth is wealth that is passed down through generations, accumulating interest over many years. This money is typically invested to increase its value rather than circulated in the economy, further impacting wealth inequalities. The other force that has affected global wealth inequalities is colonialism. Colonialism is a system through which European (and eventually American) countries exerted power over areas of the world in order to exploit their natural and human resources. Capitalism relies on the extraction of resources, laborers to process those resources, and consumers to purchase the finished products. Colonialism provided all three in the form of a global proletariat (worker) class: a group of people whose labor is the foundational resource for production. Contemporary scholars recognize colonialism as one of the most important forces in the current global system of inequality.
In the middle of the 20th century, many previously colonized countries gained independence. Due to worldwide economic downturns and ongoing colonial relationships with powerful Western countries, most did not have the means to develop their infrastructure, political organization, or economic sectors. These countries were also at a disadvantage as a result of decisions made by European powers at the Berlin Conference, which split Africa according to the wants of Western colonial powers rather than established Indigenous territories and spheres of political influence. Part of the ongoing turmoil within Africa stems from the fact that national boundaries were created with resources in mind, instead of the people who lived there.
What exactly does this have to do with social inequalities, poverty, or wealth? And how do international trade and development policies affect people without power at local levels? In the simplest terms, international structures of power affect every part of daily life for those living in poverty, especially people of color, women, and people living with disabilities. The intersections of political, economic, and social institutions reduce the number of resources available, leading to profound levels of inequity. Recognizing the long-standing effects of colonialism is vital to understanding the continuing inequities and poverty that are characteristic of so many territories that were once colonized.
To understand international structures of poverty and wealth, it is useful to also examine neocolonialism. Neocolonialism refers to the indirect ways in which modern capitalist interests continue to put pressure on poor nations through economic, political, or military means in order to further exploit wealth for multinational corporations and their allies. Rosemary Hollis, professor of Middle Eastern studies, once argued that Britain “went out the door and came back in through the window” (H.C. Foreign Affairs Committee 2013, Ev 20), meaning that it gave up its colonial holdings only to influence these nations through other means.
The main way in which neocolonialism plays out is through economic relief programs. The Global North, a term that represents powerful nations along with corporations and intergovernmental groups run by individuals from these countries, exerts power through targeted economic relief. The best-known agencies for economic relief are the World Bank Group and the International Monetary Fund (IMF). These groups, which have a lot of money, loan that money to Global South nations, which are exploited and “underdeveloped” countries that are experiencing economic or political crises. However, these loans come with many stipulations, most of which are called austerity practices . Austerity practices force governments to reduce public funding for health and education sectors, thus privatizing health care and education. For countries whose citizens are poor, introducing private health and education sectors results in a severe lack of access because many individuals cannot pay for these services.
Structural Violence
Privatization is also part of global neoliberal economics. Neoliberalism is an economic model that prioritizes privatization of public services in order to decrease government spending, based on the idea that free markets and supply and demand will lead to economic progress and development. Neoliberal policies have historically led to power structures that increase inequity for those who are already marginalized: the poor, women, and people of color. When individuals cannot fulfill their basic needs, they experience ongoing harm. Norwegian sociologist Johan Galtung calls this experience of intersecting, overlapping structures of discrimination (racism, sexism, classism, ageism, etc.) structural violence . Structural violence occurs when social institutions or practices reinforce inequalities, preventing certain social groups from obtaining basic needs. This can be an intentional or unintentional consequence.
Anthropologist and physician Paul Farmer’s (2003) work in Haiti addresses the connections between neoliberal practices and structural violence. Farmer notes that the intersection of gender, race, class, and health disparities in Haiti result in specific health challenges for which the political, economic, and social systems take little responsibility. In the township of Cange, Haiti, where residents were predominantly farmers, a dam funded by the IMF flooded a fertile valley and displaced residents from their fields, forcing them to move to the less fertile hillsides or to cities. They were provided with no subsequent public support networks, such as schools or hospitals. The amalgamation of these factors—loss of economic resources from farming, forced wage labor in the cities, and privatized education and health—resulted in what Farmer described as an inherently oppressive way of life. Many of the villagers who moved to Port-au-Prince, the capital of Haiti, were forced to rely on wage labor, with some resorting to the sex tourism industry to survive. In the 1980s, some of these villagers became infected with HIV. For these Haitians, the displacement from their villages, caused by the dam funded by the IMF, was the root cause of their later inability to fulfill basic needs and their experience of further suffering. This is a prime example of structural violence.
By understanding how class systems, poverty, wealth, and economic inequities intersect around the world, anthropologists can hope to change international programs that are based on presupposed hierarchies between the “first world” and “third world” and between the powerful and exploited classes. Anthropologist William S. Willis Jr. firmly states that “anthropologists must give no credence to the vicious theory that poor people are responsible for their poverty” (1972, 149). Theories of inequity show that poverty and success are most often the result not of individual actions but of the identities that individuals have, the diverse obstacles they have experienced, and, in large part, the lottery of their birth. Anthropological examinations of inequity must take careful consideration of institutional and structural inequalities while still upholding the ability of the individual to be an instigator of broader change. According to Willis, anthropology’s goal is to end the “poverty and powerlessness” (1972, 149) experienced globally by people of color.


5.12: Cubes in higher dimensions
This final section on elementary geometry seeks to explore fresh territory by going beyond three dimensions. Whenever we try to jump up to a new level, it can help to first take a step back and 'take a longer run up'. So please be patient if we initially take a step or two backwards.
We all know what a unit 3D-cube is. And - going backwards - it is not hard to guess what is meant by a unit “2D-cube": a unit 2D-cube is just another name for a unit square. It is then not hard to notice that a unit 3D-cube can be constructed from two unit 2D-cubes as follows:
- first position two unit 2D-cubes 1 unit apart in 3D space, with one directly above the other;
- make sure that each vertex of the lower 2D-cube is directly beneath a vertex of the upper 2D-cube;
- then join each vertex of the upper 2D-cube to the corresponding vertex below it.
Perhaps a unit 2D-cube can be constructed in a similar way from “unit lD-cubes"! This idea suggests that a unit “1D-cube” is just another name for a unit line segment .
Take the unit lD-cube to be the line segment from 0 to 1:
- position two such 1D-cubes in 2D (e.g. one joining (0,0) to (1, 0), and the other joining (0,1) to (1,1));
- check that each vertex of the lower 1D-cube is directly beneath a vertex of the upper 1D-cube;
- then join corresponding pairs of vertices - one from the upper 1D-cube and one from the lower 1D-cube ((0, 0) to (0,1), and (1,0) to (1,1)) - to obtain a unit 2D-cube.
Having taken a step back, we repeat (and reformulate) the previous construction of a 3D-cube: o
- position two such unit 2D-cubes in 3D: with one 2D-cube joining (0, 0, 0) to (1,0,0), then to (1,1, 0), then to (0,1, 0) and back to (0, 0, 0), and the other 2D-cube joining (0,0,1) to (1, 0,1), then to (1,1,1), then to (0,1,1), and back to (1,0, 0);
- with one 2D-cube directly above the other,
- then join corresponding pairs of vertices - one from the upper 2D-cube and one from the lower 2D-cube ((0,0,0) to (0,0,1), and (1, 0,0) to (1, 0,1), and (1,1, 0) to (1,1,1), and (0,1, 0) to (0,1,1)) - to obtain a unit 3D-cube.
To sum up: a unit cube in 1D, or in 2D, or in 3D:
- has as “vertices” all points whose coordinates are all “0s or 1s” (in 1D, or 2D, or 3D)
- has as “edges” all the unit segments (or unit 1D-cubes) joining vertices whose coordinates differ in exactly one place
- and a unit 3D-cube has as “faces” all the unit 2D-cubes spanned by vertices with a constant value (0 or 1) in one of the three coordinate places (that is, for the unit 3D-cube: the four vertices with x = 0, or the four vertices with x = 1; or the four vertices with y = 0, or the four vertices with y = 1; or the four vertices with z = 0, or the four vertices with z = 1).
A 3D-cube is surrounded by six 2D-cubes (or faces), and a 2D-cube is surrounded by four 1D-cubes (or faces). So it is natural to interpret the two end vertices of a 1D-cube as being '0D-cubes'. We can then see that a cube in any dimension is made up from cubes of smaller dimensions. We can also begin to make a reasonable guess as to what we might expect to find in a '4D-cuhe'.
Problem 225
(a) (i) How many vertices (i.e. 0D-cubes) are there in a 1D-cube?
(ii) How many edges (i.e. 1D-cubes) are there in a 1D-cube?
(b) (i) How many vertices (or 0D-cubes) are there in a 2D-cube?
(ii) How many “faces” (i.e. 2D-cubes) are there in a 2D-cube?
(iii) How many edges (i.e. 1D-cubes) are there in a 2D-cube?
(c) (i) How many vertices (or 0D-cubes) are there in a 3D-cube?
(ii) How many 3D-cubes are there in a 3D-cube?
(iii) How many edges (i.e. 1D-cubes) are there in a 3D-cube?
(iv) How many “faces” (i.e. 2D-cubes) are there in a 3D-cube?
(d) (i) How many vertices (or 0D-cubes) do you expect to find in a 4D-cube?
(ii) How many 4D-cubes do you expect to find in a 4D-cube?
(iii) How many edges (i.e. lD-cubes) do you expect to find in a 4D-cube?
(iv) How many “faces” (i.e. 2D-cubes) do you expect to find in a 4D-cube?
(v) How many 3D-cubes do you expect to find in a 4D-cube?
Problem 226
(a) (i) Sketch a unit 2D-cube as follows. Starting with two unit lD-cubes - one directly above the other. Then join up each vertex in the upper lD-cube to the vertex it corresponds to in the lower lD-cube (directly beneath it).
(ii) Label each vertex of your sketch with coordinates ( x, y ) ( x,y = 0 or 1) so that the lower 2D-cube has the equation “ y = 0” and the upper 2D-cube has the equation “ y = 1”.
(b) (i) Sketch a unit 3D-cube, starting with two unit 2D-cubes - one directly above the other. Then join up each vertex in the upper 2D-cube to the vertex it corresponds to in the lower 2D-cube (directly beneath it).
(ii) Label each vertex of your sketch with coordinates ( x, y, z ) (where each x,y,z = 0 or 1) so that the lower 2D-cube has the equation “ z = 0” and the upper 2D-cube has the equation “ z = 1”.
(c) (i) Now sketch a unit 4D-cube in the same way - starting with two unit 3D-cubes, one “directly above” the other.
[ Hint : In part (b) your sketch was a projection of a 3D-cube onto 2D paper, and this forced you to represent the lower and upper 2D-cubes as rhombuses rather than genuine 2D-cubes (unit squares). In part (c) you face the even more difficult task of representing a 4D-cube on 2D paper; so you must be prepared for other “distortions”. In particular, it is almost impossible to see what is going on if you try to physically position one 3D-cube “directly above” the other on 2D paper. So start with the “upper” unit 3D-cube towards the top right of your paper, and then position the “lower” unit 3D-cube not directly below it on the paper, but below and slightly to the left, before pairing off and joining up each vertex of the upper 3D-cube with the corresponding vertex in the lower 3D-cube.]
(ii) Label each vertex of your sketch with coordinates ( w, x, y, z ) (where each w,x,y, z = 0 or 1) so that the lower 3D-cube has the equation “ z = 0” and the upper 3D-cube has the equation “ z = 1”.
Problem 227 The only possible path along the edges of a 2D-cube uses each vertex once and returns to the start after visiting all four vertices.
(a) (i) Draw a path along the edges of a 3D-cube that visits each vertex exactly once and returns to the start.
(ii) Look at the sequence of coordinate triples as you follow your path. What do you notice?
(b) (i) Draw a path along the edges of a 4D-cube that visits each vertex exactly once and returns to the start.
(ii) Look at the sequence of coordinate 4-tuples as you follow your path. What do you notice?


12: DNA and RNA
Children inherit some characteristics from each parent. Siblings typically look similar to each other, but not exactly the same—except in the case of identical twins. How can we explain these phenomena? The answers lie in heredity (the transmission of traits from one generation to the next) and genetics (the science of heredity). Because humans reproduce sexually, 50% of a child’s genes come from the mother’s egg cell and the remaining 50% from the father’s sperm cell. Sperm and egg are formed through the process of meiosis , where DNA recombination occurs. Thus, there is no predictable pattern as to which 50% comes from which parent. Thus, siblings have only some genes, and their associated characteristics, in common. Identical twins are the exception, because they are genetically identical.
Genetic differences among related microbes also dictate many observed biochemical and virulence differences. For example, some strains of the bacterium Escherichia coli are harmless members of the normal microbiota in the human gastrointestinal tract. Other strains of the same species have genes that give them the ability to cause disease. In bacteria, such genes are not inherited via sexual reproduction, as in humans. Often, they are transferred via plasmids, small circular pieces of double-stranded DNA that can be exchanged between prokaryotes.
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- 12.1: Using Microbiology to Discover the Secrets of Life
- DNA was discovered and characterized long before its role in heredity was understood. Microbiologists played significant roles in demonstrating that DNA is the hereditary information found within cells. In the 1850s and 1860s, Gregor Mendel experimented with true-breeding garden peas to demonstrate the heritability of specific observable traits. In 1869, Friedrich Miescher isolated and purified a compound rich in phosphorus from the nuclei of white blood cells; he named the compound nuclein.
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- 12.2: Structure and Function of DNA
- Nucleic acids are composed of nucleotides, each of which contains a pentose sugar, a phosphate group, and a nitrogenous base. Deoxyribonucleotides within DNA contain deoxyribose as the pentose sugar. DNA contains the pyrimidines cytosine and thymine, and the purines adenine and guanine. Nucleotides are linked together by phosphodiester bonds between the 5ʹ phosphate group of one nucleotide and the 3ʹ hydroxyl group of another.
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- 12.3: Structure and Function of RNA
- Ribonucleic acid (RNA) is typically single stranded and contains ribose as its pentose sugar and the pyrimidine uracil instead of thymine. An RNA strand can undergo significant intramolecular base pairing to take on a three-dimensional structure. There are three main types of RNA, all involved in protein synthesis. Messenger RNA (mRNA) serves as the intermediary between DNA and the synthesis of protein products during translation.
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- 12.4: The Structure and Function of Cellular Genomes
- The entire genetic content of a cell is its genome. Genes code for proteins, or stable RNA molecules, each of which carries out a specific function in the cell. Although the genotype that a cell possesses remains constant, expression of genes is dependent on environmental conditions. A phenotype is the observable characteristics of a cell (or organism) at a given point in time and results from the complement of genes currently being used.
Thumbnail: In the laboratory, the double helix can be denatured to single-stranded DNA through exposure to heat or chemicals, and then renatured through cooling or removal of chemical denaturants to allow the DNA strands to reanneal. (credit: modification of work by Hernández-Lemus E, Nicasio-Collazo LA, Castañeda-Priego R)


2.3: Plan for successful teamwork
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Tuckman suggested that teams move through stages in their life cycle: forming, storming, norming, and performing. At each stage, the group will work through a series of interpersonal tasks, as well as a series of project-related tasks.
In the first section of this module, you explored the components of a Community of Inquiry. Both cognitive presence and social presence are required in the online learning community. Tuckman’s model of team development also indicates that both components are needed. In a class-based team, it may be easy to focus only on the cognitive output of the group — the creation of the project, paper, or presentation. However, as you can observe from Tuckman’s model, a well-functioning team requires its members to exhibit social presence throughout, communicating well in interpersonal interactions.
In the days ahead, you will likely find yourself on a newly forming team in an online environment. Consider the strategies you plan to use to demonstrate social presence and form a strong interpersonal foundation for your newly forming team. [1]


1.4.5: Dividing Radical Expressions
By the end of this section, you will be able to:
- Divide radical expressions
- Rationalize a one term denominator
- Rationalize a two term denominator
Before you get started, take this readiness quiz.
- Simplify \(\dfrac{30}{48}\).
- Simplify \(x^{2}⋅x^{4}\).
- Multiply \((7+3x)(7−3x)\).
Divide Radical Expressions
We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.
If \(\sqrt{a}\) and \(\sqrt{b}\) are real numbers with \(b≠0\), then
\(\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}},\)
or, in the language of exponents:
\(\left(\dfrac{a}{b}\right)^\frac12=\dfrac{a^\frac12}{b^\frac12}.\)
We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.
Simplify \(\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\).
- Solution
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\(\quad\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\) Rewrite using the quotient property, \(=\sqrt{\dfrac{72 x^{3}}{162 x}}\) Remove common factors. \(=\sqrt{\dfrac{\cancel{18}\cdot 4\cdot x^2\cdot\cancel{x}}{\cancel{18}\cdot 9\cdot\cancel{x}}}\) Simplify. \(=\sqrt{\dfrac{4 x^{2}}{9}}\) Simplify the radical. \(=\dfrac{2 x}{3}\)
Simplify \(\dfrac{\sqrt{50 s^{3}}}{\sqrt{128 s}}\).
- Answer
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\(\dfrac{5s}{8}\)
Simplify \(\dfrac{\sqrt{75 q^{5}}}{\sqrt{108 q}}\).
- Answer
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\(\dfrac{5 q^{2}}{6}\)
Simplify \(\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\).
- Solution
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\(\quad\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\) Rewrite using the quotient property. \(=\sqrt{\dfrac{147 a b^{8}}{3 a^{3} b^{4}}}\) Remove common factors in the fraction. \(=\sqrt{\dfrac{49 b^{4}}{a^{2}}}\) Simplify the radical. \(=\dfrac{7 b^{2}}{a}\)
Simplify \(\dfrac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}\).
- Answer
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\(\dfrac{9 x^{2}}{y^{2}}\)
Simplify \(\dfrac{\sqrt{300 m^{3} n^{7}}}{\sqrt{3 m^{5} n}}\).
- Answer
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\(\dfrac{10 n^{3}}{m}\)
Simplify \(\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\).
- Solution
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\(\quad\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\) Rewrite using the quotient property. \(=\sqrt{\dfrac{54 x^{5} y^{3}}{3 x^{2} y}}\) Remove common factors in the fraction. \(=\sqrt{18 x^{3} y^{2}}\) Rewrite the radicand as a product using the largest perfect square factor. \(=\sqrt{9 x^{2} y^{2} \cdot 2 x}\) Rewrite the radical as the product of two radicals. \(=\sqrt{9 x^{2} y^{2}} \cdot \sqrt{2 x}\) Simplify. \(=3 x y \sqrt{2 x}\)
Simplify \(\dfrac{\sqrt{64 x^{4} y^{5}}}{\sqrt{2 x y^{3}}}\).
- Answer
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\(4 x y \sqrt{2 x}\)
Simplify \(\dfrac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}}\).
- Answer
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\(4 a b \sqrt{3 b}\)
Rationalize a One-Term Denominator
Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process! For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator. This process is still used today, and is useful in other areas of mathematics too. In particular, when such expressions will be added or further simplified it is handy to have them in a form where there are no radicals in the denominator.
Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator should still be rationalized. It is not considered simplified if the denominator contains a radical.
Similarly, a radical expression is not considered simplified if the radicand contains a fraction.
Simplified Radical Expressions
A radical expression is considered simplified if there are
- no factors in the radicand have perfect powers of the index
- no fractions in the radicand
- no radicals in the denominator of a fraction
To rationalize a denominator with a square root, we use the property that \(\left(\sqrt{a}\right)^{2}=a\). If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Simplify:
a. \(\dfrac{4}{\sqrt{3}}\)
b. \(\sqrt{\dfrac{3}{20}}\)
c. \(\dfrac{3}{\sqrt{6 x}}\)
- Solution
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To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.
a.
\(\quad\dfrac{4}{\sqrt{3}}\) Multiply both the numerator and denominator by \(\sqrt{3}\). \(=\dfrac{4\cdot {\color{red}{\sqrt 3}}}{\sqrt{3}\cdot {\color{red}{\sqrt 3}}}\) Simplify. \(=\dfrac{4\sqrt{3}}{3}\) b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.
\(\quad\sqrt{\dfrac{3}{20}}\)
The fraction is not a perfect square, so rewrite using the Quotient Property. \(=\dfrac{\sqrt{3}}{\sqrt{20}}\)
Simplify the denominator. \(=\dfrac{\sqrt{3}}{2\sqrt{5}}\) Multiply the numerator and denominator by \(\sqrt{5}\). \(=\dfrac{\sqrt{3}\cdot{\color{red}{\sqrt{5}}}}{2\sqrt{5}\cdot{\color{red}{\sqrt{5}}}}\) Simplify. \(=\dfrac{\sqrt{15}}{2\cdot 5}\) Simplify. \(=\dfrac{\sqrt{15}}{10}\) c.
\(\quad\dfrac{3}{\sqrt{6 x}}\) Multiply the numerator and denominator by \(\sqrt{6x}\). \(=\dfrac{3\cdot{\color{red}{\sqrt{6x}}}}{\sqrt{6 x}\cdot {\color{red}{\sqrt{6x}}}}\)
Simplify. \(=\dfrac{3\sqrt{6x}}{6x}\)
Simplify. \(=\dfrac{\sqrt{6x}}{2x}\)
Simplify:
a. \(\dfrac{5}{\sqrt{3}}\)
b. \(\sqrt{\dfrac{3}{32}}\)
c. \(\dfrac{2}{\sqrt{2 x}}\)
- Answer
-
a. \(\dfrac{5 \sqrt{3}}{3}\)
b. \(\dfrac{\sqrt{6}}{8}\)
c. \(\dfrac{\sqrt{2 x}}{x}\)
Simplify:
a. \(\dfrac{6}{\sqrt{5}}\)
b. \(\sqrt{\dfrac{7}{18}}\)
c. \(\dfrac{5}{\sqrt{5 x}}\)
- Answer
-
a. \(\dfrac{6 \sqrt{5}}{5}\)
b. \(\dfrac{\sqrt{14}}{6}\)
c. \(\dfrac{\sqrt{5 x}}{x}\)
When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.
Rationalize a Two-Term Denominator
When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates Pattern to rationalize the denominator .
\(\begin{array}{l l}{\quad (a-b)(a+b)} & \qquad \qquad {(2-\sqrt{5})(2+\sqrt{5})} \\ {= a^{2}-b^{2}} &{ = 2^{2}-(\sqrt{5})^{2}} \\ {}&{= 4-5} \\ {}&{=-1}\end{array}\)
When we multiple a binomial that includes a square root by its conjugate, the product has no square roots.
Simplify \(\dfrac{5}{2-\sqrt{3}}\).
- Solution
-
\(\quad\dfrac{5}{2-\sqrt{3}}\) Multiply the numerator and denominator by the conjugate of the denominator. \(=\dfrac{5{\color{red}{\;(2+\sqrt{3})}}}{\left(2-\sqrt{3}\right){\color{red}{\left(2+\sqrt{3}\right)}}}\) Multiply the conjugates in the denominator. \(=\dfrac{5\left(2+\sqrt{3}\right)}{2^2-\left(\sqrt{3}\right)^2}\) Simplify the denominator. \(=\dfrac{5\left(2+\sqrt{3}\right)}{4-3}\) Simplify the denominator. \(=\dfrac{5\left(2+\sqrt{3}\right)}{1}\) Simplify. \(=5\left(2+\sqrt{3}\right)\)
Simplify \(\dfrac{3}{1-\sqrt{5}}\).
- Answer
-
\(-\dfrac{3(1+\sqrt{5})}{4}\)
Simplify \(\dfrac{2}{4-\sqrt{6}}\).
- Answer
-
\(\dfrac{4+\sqrt{6}}{5}\)
Notice we did not distribute the \(5\) in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.
Simplify \(\dfrac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\).
- Solution
-
\(\quad\dfrac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\) Multiply the numerator and denominator by the conjugate of the denominator. \(=\dfrac{\sqrt{3}{\color{red}{\left(\sqrt{u}+\sqrt{6}\right)}}}{\left(\sqrt{u}-\sqrt{6}\right){\color{red}{\left(\sqrt{u}+\sqrt{6}\right)}}}\) Multiply the conjugates in the denominator. \(=\dfrac{\sqrt{3}\left(\sqrt{u}+\sqrt{6}\right)}{\left(\sqrt{u}\right)^2-\left(\sqrt{6}\right)^2}\) Simplify the denominator. \(=\dfrac{\sqrt{3}\left(\sqrt{u}+\sqrt{6}\right)}{u-6}\)
Simplify \(\dfrac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}\).
- Answer
-
\(\dfrac{\sqrt{5}(\sqrt{x}-\sqrt{2})}{x-2}\)
Simplify \(\dfrac{\sqrt{10}}{\sqrt{y}-\sqrt{3}}\).
- Answer
-
\(\dfrac{\sqrt{10}(\sqrt{y}+\sqrt{3})}{y-3}\)
Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.
Simplify \(\dfrac{\sqrt{x}+\sqrt{7}}{\sqrt{x}-\sqrt{7}}\).
- Solution
-
\(\quad\dfrac{\sqrt{x}+\sqrt{7}}{\sqrt{x}-\sqrt{7}}\) Multiply the numerator and denominator by the conjugate of the denominator. \(=\dfrac{\left(\sqrt{x}+\sqrt{7}\right){\color{red}{\left(\sqrt{x}+\sqrt{7}\right)}}}{\left(\sqrt{x}-\sqrt{7}\right){\color{red}{\left(\sqrt{x}+\sqrt{7}\right)}}}\) Multiply the conjugates in the denominator. \(=\dfrac{\left(\sqrt{x}+\sqrt{7}\right)\left(\sqrt{x}+\sqrt{7}\right)}{\left(\sqrt{x}\right)^2-\left(\sqrt{7}\right)^2}\) Simplify the denominator. \(=\dfrac{\left(\sqrt{x}+\sqrt{7}\right)^2}{x-7}\) We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.
Simplify \(\dfrac{\sqrt{p}+\sqrt{2}}{\sqrt{p}-\sqrt{2}}\).
- Answer
-
\(\dfrac{(\sqrt{p}+\sqrt{2})^{2}}{p-2}\)
Simplify \(\dfrac{\sqrt{q}-\sqrt{10}}{\sqrt{q}+\sqrt{10}}\).
- Answer
-
\(\dfrac{(\sqrt{q}-\sqrt{10})^{2}}{q-10}\)
- Explain why the process of rationalizing the denominator described above works.
- What is one reason you may want to rationalize the denominator?
- Why does the ‘conjugate’ play a role in accomplishing this?
Simplify:
a. \(\dfrac{3-3\sqrt{3a}}{4\sqrt{8a}}\)
b. \(\dfrac{\sqrt{5}+3}{4-\sqrt{5}}\)
Key Concepts
-
Quotient Property of Radical Expressions
- If \(\sqrt{a}\) and \(\sqrt{b}\) are real numbers, \(b≠0\), and then, \(\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\).
-
Simplified Radical Expressions
-
A radical expression is considered simplified if there are:
- no factors in the radicand that have perfect powers of the index
- no fractions in the radicand
- no radicals in the denominator of a fraction
-
A radical expression is considered simplified if there are:


1: DNA, Genes, Chromosomes and Genome
The molecular foundation of all living organisms lies in DNA — deoxyribonucleic acid — a remarkable molecule that encodes the instructions needed for growth, development, and reproduction. DNA is organized into genes , the functional units of heredity, which guide the production of proteins that determine the structure and function of cells. These genes are meticulously arranged on chromosomes , thread-like structures within the cell nucleus that ensure accurate distribution of genetic information during cell division.
Collectively, an organism's complete set of DNA, including all its genes, is known as its genome . The human genome, for instance, spans over 3 billion base pairs and contains about 20,000-25,000 genes distributed across 23 pairs of chromosomes. Understanding the interplay between DNA, genes, chromosomes, and the genome is crucial for exploring the complexity of genetic inheritance, disease mechanisms, and the diversity of life. This chapter delves into the intricate architecture and function of these genetic elements, providing a foundation for the study of molecular biology and genetics.


12.1C: Sensory Modalities
A sensory modality (also called a stimulus modality) is an aspect of a stimulus or what is perceived after a stimulus.
- Describe the sensory modalities of the peripheral nervous system
Key Points
- The basic sensory modalities include: light, sound, taste, temperature, pressure, and smell.
- A broadly acceptable definition of a sense is: A system that consists of a group of sensory cell types, responding to a specific physical phenomenon, and corresponding to a particular group of regions within the brain where the signals are received and interpreted.
- Multimodal perception is the ability of the mammalian nervous system to combine different inputs of the sensory system. Nociception (physiological pain ) signals nerve damage or damage to tissue. The three types of pain receptors are cutaneous (skin), somatic (joints and bones), and visceral (body organs ).
- Proprioception, the kinesthetic sense, provides the parietal cortex of the brain with information on the relative positions of the parts of the body.
Key Terms
- thermoreception : A physiological response to relative or absolute changes in temperature.
- modality : Also known as stimulus modality, it is one feature of a complex stimulus; for example, temperature, pressure, sound, or taste.
- utricle : Stimulates hair cells of the inner ear to detect motion and orientation.
- saccule : A bed of sensory cells situated in the inner ear that translates head movements into neural impulses that the brain can interpret.
- circadian : Any biological process that displays an endogenous, entrainable oscillation of about 24 hours.
- ultradian : A recurrent period or cycle repeated throughout a 24-hour circadian day.
- mechanoreception : A physiological response to mechanical forces like pressure, touch, and vibration.
- bipolar cell : Specialized sensory neuron for the transmission of special senses.
Sensing
Senses are transducers from the physical world to the realm of the mind. Another broadly acceptable definition of a sense is: A system that consists of a group of sensory cell types, responding to a specific physical phenomenon, and corresponding to a particular group of regions within the brain where the signals are received and interpreted.
Disputes about the number of senses typically arise around classification of the various cell types, and their mapping to regions of the brain.
Sensory Modalities
A sensory modality (also called a stimulus modality) is an aspect of a stimulus or what is perceived after a stimulus. The term sensory modality is often used interchangeably with sense. The basic sensory modalities include: light, sound, taste, temperature, pressure, and smell.
Light Modality
The sensory modality for vision is light. To perceive a light stimulus, the eye must first refract the light so that it directly hits the retina. The transduction of light into neural activity occurs via the photoreceptors in the retina.
When a particle of light hits the photoreceptors of the eye, the photopigment of the photoreceptor undergoes a chemical change leading to a chain of chemical reactions occur. A message is sent to a neuron called the bipolar cell through the use of a nerve impulse. Finally, a message is sent to the ganglion cell and then, finally, the brain.
Sound Modality
The sensory modality for audition is sound. Sound is created through air pressure. A vibrating object compresses the surrounding molecules of air as it moves towards a given point, and expands the molecules as it moves away from the point.
The eardrum is stimulated by vibrations in the air. It collects and sends these vibrations to receptor cells. The ossicles (three tiny bones in the middle ear) pass the vibrations to the fluid-filled cochlea (a spiral, shell-shaped auditory organ of the inner ear ). The vibrations move through the liquid in the cochlea where the receptive organ is able to sense it.
Taste Modality
Taste stimuli are encountered by receptor cells located in taste buds on the tongue and pharynx. Receptor cells disseminate onto different neurons and convey the message of a particular taste in a single medullar nucleus.
Taste perception is created by combining multiple sensory inputs. Different modalities help determine the perception of taste.
Temperature Modality
Temperature modality excites or elicits a symptom through cold or hot temperature. The cutaneous somatosensory system detects changes in temperature.
Thermal stimuli from a homeostatic set point excite temperature specific sensory nerves in the skin. Specific thermosensory fibers respond to warmth and to cold.
Pressure Modality
Tactile stimulation can be direct, such as through bodily contact, or indirect, such as through the use of a tool or probe. Tactual perception gives information regarding cutaneous stimuli (pressure, vibration, and temperature), kinesthetic stimuli (limb movement), and proprioceptive stimuli (position of the body).
Smell Modality
The sense of smell is called olfaction. Materials constantly shed molecules, which float into the nose or are taken in through breathing. Inside the nasal chambers is the neuroepithelium lining.
It contains the receptors responsible for detecting molecules that are small enough to smell. These receptor neurons then synapse at the olfactory cranial nerve, which sends the information to the olfactory bulbs in the brain for initial processing.
Multimodal Perception
Multimodal perception is the ability of the mammalian nervous system to combine all of the different inputs of the sensory system to result in an enhanced detection or identification of a particular stimulus.
Integration of all sensory modalities occurs when multimodal neurons receive sensory information that overlaps with different modalities. Multimodal perception comes into effect when a unimodal stimulus fails to produce a response.
Multisensory perception : This is a diagram of how multimodal perception is created by the overlapping and combining of different inputs from the sensory systems.
Additional Senses
Balance (or equilibrioception) is the sense that allows an organism to sense body movement, direction, and acceleration, and also attain and maintain postural equilibrium and balance. The organ of equilibrioception is the vestibular labyrinthine system found in both of the inner ears.
In technical terms, this organ is responsible for two senses: angular momentum and acceleration (known together as equilibrioception). The vestibular nerve conducts information from sensory receptors in three ampulla, each of which sense fluid motion in three semicircular canals caused by a three-dimensional rotation of the head.
The vestibular nerve also conducts information from the utricle and the saccule; these contain hair-like sensory receptors that bend under the weight of otoliths (small crystals of calcium carbonate) that provide the inertia needed to detect head rotation, linear acceleration, and the direction of gravitational force.
Inner ear : Inner ear anatomy showing utricle, saccule, and vestibular nerve.
Thermoception is the sense of heat or absence of heat (cold) by the skin and internal skin passages. Perceiving changes in temperature is referred to as heat flux (the rate of heat flow) in these areas.
There are specialized receptors for cold (declining temperature) and heat. The cold receptors infer wind direction, an important part in the animal’s sense of smell. The heat receptors are sensitive to infrared radiation and can occur in specialized organs, for instance in pit vipers.
The thermoceptors in the skin are quite different from the homeostatic thermoceptors in the brain (hypothalamus), which provides feedback on internal body temperature.
Proprioception, the kinesthetic sense, provides the parietal cortex of the brain with information on the relative positions of the parts of the body. Neurologists test this sense by telling patients to close their eyes and touch their own nose with the tip of a finger. Assuming proper proprioceptive function, at no time will the person lose awareness of where their hand actually is, even though it is not being detected by any of the other senses. Proprioception and touch are related in subtle ways, and their impairment results in deep and surprising deficits in perception and action.
Nociception (physiological pain) signals nerve or other tissue damage. The three types of pain receptors are cutaneous (skin), somatic (joints and bones), and visceral (body organs).
It was previously believed that pain was simply the overloading of pressure receptors, but research in the first half of the 20th century showed that pain is a distinct phenomenon that intertwines with all of the other senses, including touch.
Chronoception refers to how the passage of time is perceived and experienced. Although the sense of time is not associated with a specific sensory system, psychological and neuro-scientific research indicates that human brains do have a system governing the perception of time.
It is composed of a highly distributed system involving the cerebral cortex, cerebellum and basal ganglia. One particular component, the suprachiasmatic nucleus, is responsible for the circadian (daily) rhythm, while other cell clusters appear capable of shorter-range (ultradian) timekeeping.
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Chapter 3: Criminal Law
Learning Objectives
After reading this section, students will be able to:
- Distinguish between a criminal wrong, a civil wrong, and moral wrong.
- Identify the many ways in which criminal law is classified.
- Recognize the many sources of substantive and procedural criminal law.
- Identify the limitations that the federal constitution and state constitutions place on creating substantive laws and enforcing those laws.
- Recognize the importance of rule of law in American jurisprudence and understand the importance of judicial review in achieving rule of law.
This section examines the fundamental principles of criminal law. It describes the functions of formal criminal law (what criminal law does and what it cannot do), how crimes differ from civil and moral wrongs, and various classification schemes used in discussing criminal law. This section also examines the sources of substantive and procedural criminal law (where we look to find our criminal law), the limitations that the constitution places on both substantive criminal law and procedural criminal law, and the important concept of the rule of law in American jurisprudence (legal theory).
Critical Thinking Questions
- What does formal law do well? What does formal law not do so well?
- Should we be able to impose sanctions for violations of moral wrongs?
- Consider the constitutional requirement of separate but equal branches of government. Why do you think the drafters of the constitution intended each of the branches of government to be a check on each other? How does that “play out” when deciding what laws should be made and what laws should be enforced? What current issues are you aware of that highlight the importance of three separate but equal branches of government?
- How does direct democracy (in the form of ballot measures and propositions) influence substantive criminal law (creating crimes and punishing crimes). What, if any, are the advantages of using direct democracy to create and punish crime? What, if any, are the disadvantages?
- Consider state-wide decriminalization of marijuana possession and use across the nation and the federal statute banning possession and use of marijuana. How should this federal/state conflict be resolved? Does your opinion change if the behavior is one that you favor or disfavor?
-
- 3.6: Classifications of Law
- Classification schemes allow us to discuss aspects or characteristics of the criminal law. Some classifications have legal significance, meaning that how a crime is classified may make a difference in how the case is processed or what type of punishment can be imposed. Some classifications historically mattered (had legal significance), but no longer have much consequence. Finally, some classifications have no legal significance, meaning the classification exists only to help us organize laws.
-
- 3.7: Substantive Law - Defining Crimes, Inchoate Liability, Accomplice Liability, and Defenses
- Substantive law includes laws that define crime, meaning laws that tell us what elements the government needs to prove in order to establish that this crime has been committed. Substantive law also includes the definitions of inchoate crimes (incomplete crimes) of conspiracies, solicitations, and attempts. Substantive law also sets forth accomplice liability (when a person will be held responsible when they work in concert with others to complete a crime).
-
- 3.12: Procedural Law
- Procedural law governs the process used to investigate and prosecute individuals who commits a crime. Procedural law also governs the ways people convicted of a crime may challenge their convictions. The source of procedural law includes the same sources of law that govern substantive criminal law: the constitution, cases law or judicial opinions, statutes, and common law. Most substantive criminal law is now statutory and most procedural law is found in judicial opinions.


23.5: Pulsars and the Discovery of Neutron Stars
By the end of this section, you will be able to:
- Explain the research method that led to the discovery of neutron stars, located hundreds or thousands of light-years away
- Describe the features of a neutron star that allow it to be detected as a pulsar
- List the observational evidence that links pulsars and neutron stars to supernovae
After a type II supernova explosion fades away, all that is left behind is either a neutron star or something even more strange, a black hole . We will describe the properties of black holes in Black Holes and Curved Spacetime, but for now, we want to examine how the neutron stars we discussed earlier might become observable.
Neutron stars are the densest objects in the universe; the force of gravity at their surface is 10 11 times greater than what we experience at Earth’s surface. The interior of a neutron star is composed of about 95% neutrons, with a small number of protons and electrons mixed in.  Its diameter is more like the size of a small town or an asteroid than a star. (Table 23.3 compares the properties of neutron stars and white dwarfs .) Because it is so small, a neutron star probably strikes you as the object least likely to be observed from thousands of light-years away. Yet neutron stars do manage to signal their presence across vast gulfs of space.
| Property | White Dwarf | Neutron Star |
|---|---|---|
| Mass (Sun = 1) | 0.6 (always <1.4) | Always >1.4 and <3 |
| Radius | 7000 km | 10 km |
| Density | g/cm 3 | 10 14 g/cm 3 |
The Discovery of Neutron Stars
In 1967, Jocelyn Bell , a research student at Cambridge University, was studying distant radio sources with a special detector that had been designed and built by her advisor Antony Hewish to find rapid variations in radio signals. The project computers spewed out reams of paper showing where the telescope had surveyed the sky, and it was the job of Hewish’s graduate students to go through it all, searching for interesting phenomena. In September 1967, Bell discovered what she called “a bit of scruff”—a strange radio signal unlike anything seen before.
What Bell had found, in the constellation of Vulpecula, was a source of rapid, sharp, intense, and extremely regular pulses of radio radiation. Like the regular ticking of a clock, the pulses arrived precisely every 1.33728 seconds. Such exactness first led the scientists to speculate that perhaps they had found signals from an intelligent civilization. Radio astronomers even half-jokingly dubbed the source “LGM” for “little green men.” Soon, however, three similar sources were discovered in widely separated directions in the sky.
When it became apparent that this type of radio source was fairly common, astronomers concluded that they were highly unlikely to be signals from other civilizations. By today, nearly 3000 such sources have been discovered; they are now called pulsars , short for “pulsating radio sources.”
The pulse periods of different pulsars range from a little longer than 1/1000 of a second to nearly 10 seconds. At first, the pulsars seemed particularly mysterious because nothing could be seen at their location on visible-light photographs. But then a pulsar was discovered right in the center of the Crab Nebula , a cloud of gas produced by SN 1054 , a supernova that was recorded by the Chinese in 1054 (Figure 23.14). The energy from the Crab Nebula pulsar arrives in sharp bursts that occur 30 times each second—with a regularity that would be the envy of a Swiss watchmaker. In addition to pulses of radio energy, we can observe pulses of visible light and X-rays from the Crab Nebula. The fact that the pulsar was just in the region of the supernova remnant where we expect the leftover neutron star to be immediately alerted astronomers that pulsars might be connected with these elusive “corpses” of massive stars.
The Crab Nebula is a fascinating object. The whole nebula glows with radiation at many wavelengths, and its overall energy output is more than 100,000 times that of the Sun—not a bad trick for the remnant of a supernova that exploded almost a thousand years ago. Astronomers soon began to look for a connection between the pulsar and the large energy output of the surrounding nebula.
Read the transcript of an interesting interview with Jocelyn Bell (Burnell) to learn about her life and work (this is part of a project at the American Institute of Physics to record interviews with pathbreaking scientists while they are still alive).
A Spinning Lighthouse Model
By applying a combination of theory and observation, astronomers eventually concluded that pulsars must be spinning neutron stars . According to this model, a neutron star is something like a lighthouse on a rocky coast (Figure 23.15). To warn ships in all directions and yet not cost too much to operate, the light in a modern lighthouse turns, sweeping its beam across the dark sea. From the vantage point of a ship, you see a pulse of light each time the beam points in your direction. In the same way, radiation from a small region on a neutron star sweeps across the oceans of space, giving us a pulse of radiation each time the beam points toward Earth.
Neutron stars are ideal candidates for such a job because the collapse has made them so small that they can turn very rapidly. Recall the principle of the conservation of angular momentum from Newton’s Great Synthesis: if an object gets smaller, it can spin more rapidly. Even if the parent star was rotating very slowly when it was on the main sequence, its rotation had to speed up as it collapsed to form a neutron star. With a diameter of only 10 to 20 kilometers, a neutron star can complete one full spin in only a fraction of a second. This is just the sort of time period we observe between pulsar pulses.
Any magnetic field that existed in the original star will be highly compressed when the core collapses to a neutron star. At the surface of the neutron star, in the outer layer consisting of ordinary matter (and not just pure neutrons), protons and electrons are caught up in this spinning field and accelerated nearly to the speed of light. In only two places—the north and south magnetic poles—can the trapped particles escape the strong hold of the magnetic field (Figure 23.16). The same effect can be seen (in reverse) on Earth, where charged particles from space are kept out by our planet’s magnetic field everywhere except near the poles. As a result, Earth’s auroras (caused when charged particles hit the atmosphere at high speed) are seen mainly near the poles.
Note that in a neutron star, the magnetic north and south poles do not have to be anywhere close to the north and south poles defined by the star’s rotation. In the same way, we discussed in the chapter on The Giant Planets that the magnetic poles on the planets Uranus and Neptune are not lined up with the poles of the planet’s spin. Figure 23.16 shows the poles of the magnetic field perpendicular to the poles of rotation, but the two kinds of poles could make any angle.
In fact, the misalignment of the rotational axis with the magnetic axis plays a crucial role in the generation of the observed pulses in this model. At the two magnetic poles, the particles from the neutron star are focused into a narrow beam and come streaming out of the whirling magnetic region at enormous speeds. They emit energy over a broad range of the electromagnetic spectrum. The radiation itself is also confined to a narrow beam, which explains why the pulsar acts like a lighthouse. As the rotation carries first one and then the other magnetic pole of the star into our view, we see a pulse of radiation each time.
Tests of the Model
This explanation of pulsars in terms of beams of radiation from highly magnetic and rapidly spinning neutron stars is a very clever idea. But what evidence do we have that it is the correct model? First, we can measure the masses of some pulsars, and they do turn out be in the range of 1.4 to 1.8 times that of the Sun—just what theorists predict for neutron stars. The masses are found using Kepler’s law for those few pulsars that are members of binary star systems.
But there is an even-better confirming argument, which brings us back to the Crab Nebula and its vast energy output. When the high-energy charged particles from the neutron star pulsar hit the slower-moving material from the supernova, they energize this material and cause it to “glow” at many different wavelengths—just what we observe from the Crab Nebula. The pulsar beams are a power source that “light up” the nebula long after the initial explosion of the star that made it.
Who “pays the bills” for all the energy we see coming out of a remnant like the Crab Nebula? After all, when energy emerges from one place, it must be depleted in another. The ultimate energy source in our model is the rotation of the neutron star, which propels charged particles outward and spins its magnetic field at enormous speeds. As its rotational energy is used to excite the Crab Nebula year after year, the pulsar inside the nebula slows down. As it slows, the pulses come a little less often; more time elapses before the slower neutron star brings its beam back around.
Several decades of careful observations have now shown that the Crab Nebula pulsar is not a perfectly regular clock as we originally thought: instead, it is gradually slowing down. Having measured how much the pulsar is slowing down, we can calculate how much rotation energy the neutron star is losing. Remember that it is very densely packed and spins amazingly quickly. Even a tiny slowing down can mean an immense loss of energy.
To the satisfaction of astronomers, the rotational energy lost by the pulsar turns out to be the same as the amount of energy emerging from the nebula surrounding it. In other words, the slowing down of a rotating neutron star can explain precisely why the Crab Nebula is glowing with the amount of energy we observe.
The Evolution of Pulsars
From observations of the pulsar s discovered so far, astronomers have concluded that one new pulsar is born somewhere in the Galaxy every 25 to 100 years, the same rate at which supernovae are estimated to occur. Calculations suggest that the typical lifetime of a pulsar is about 10 million years; after that, the neutron star no longer rotates fast enough to produce significant beams of particles and energy, and is no longer observable. We estimate that there are about 100 million neutron stars in our Galaxy, most of them rotating too slowly to come to our notice.
The Crab pulsar is rather young (only about 960 years old) and has a short period, whereas other, older pulsars have already slowed to longer periods. Pulsars thousands of years old have lost too much energy to emit appreciably in the visible and X-ray wavelengths, and they are observed only as radio pulsars; their periods are a second or longer.
There is one other reason we can see only a fraction of the pulsars in the Galaxy. Consider our lighthouse model again. On Earth, all ships approach on the same plane—the surface of the ocean—so the lighthouse can be built to sweep its beam over that surface. But in space, objects can be anywhere in three dimensions. As a given pulsar’s beam sweeps over a circle in space, there is absolutely no guarantee that this circle will include the direction of Earth. In fact, if you think about it, many more circles in space will not include Earth than will include it. Thus, we estimate that we are unable to observe a large number of neutron stars because their pulsar beams miss us entirely.
At the same time, it turns out that only a few of the pulsars discovered so far are embedded in the visible clouds of gas that mark the remnant of a supernova. This might at first seem mysterious, since we know that supernovae give rise to neutron stars and we should expect each pulsar to have begun its life in a supernova explosion. But the lifetime of a pulsar turns out to be about 100 times longer than the length of time required for the expanding gas of a supernova remnant to disperse into interstellar space. Thus, most pulsars are found with no other trace left of the explosion that produced them.
In addition, some pulsars are ejected by a supernova explosion that is not the same in all directions. If the supernova explosion is stronger on one side, it can kick the pulsar entirely out of the supernova remnant (some astronomers call this “getting a birth kick”). We know such kicks happen because we see a number of young supernova remnants in nearby galaxies where the pulsar is to one side of the remnant and racing away at several hundred miles per second (Figure 23.17).
On December 27, 2004, Earth was bathed with a stream of X-ray and gamma-ray radiation from a neutron star known as SGR 1806-20. What made this event so remarkable was that, despite the distance of the source, its tidal wave of radiation had measurable effects on Earth’s atmosphere. The apparent brightness of this gamma-ray flare was greater than any historical star explosion.
The primary effect of the radiation was on a layer high in Earth’s atmosphere called the ionosphere . At night, the ionosphere is normally at a height of about 85 kilometers, but during the day, energy from the Sun ionizes more molecules and lowers the boundary of the ionosphere to a height of about 60 kilometers. The pulse of X-ray and gamma-ray radiation produced about the same level of ionization as the daytime Sun. It also caused some sensitive satellites above the atmosphere to shut down their electronics.
Measurements by telescopes in space indicate that SGR 1806-20 was a special type of fast-spinning neutron star called a magnetar . Astronomers Robert Duncan and Christopher Thomson gave them this name because their magnetic fields are stronger than that of any other type of astronomical source—in this case, about 800 trillion times stronger than the magnetic field of Earth.
A magnetar is thought to consist of a superdense core of neutrons surrounded by a rigid crust of atoms about a mile deep with a surface made of iron. The magnetar’s field is so strong that it creates huge stresses inside that can sometimes crack open the hard crust, causing a starquarke. The vibrating crust produces an enormous blast of radiation. An astronaut 0.1 light-year from this particular magnetar would have received a fatal does from the blast in less than a second.
Fortunately, we were far enough away from magnetar SGR 1806-20 to be safe. Could a magnetar ever present a real danger to Earth? To produce enough energy to disrupt the ozone layer, a magnetar would have to be located within the cloud of comets that surround the solar system, and we know no magnetars are that close. Nevertheless, it is a fascinating discovery that events on distant star corpses can have measurable effects on Earth.


Intro to Phases and Intermolecular Forces
Skills to Develop
- Describe phases and phase changes on a molecular level
Phases of matter mean the state of a material, like solid, liquid or gas. How can we predict whether a material will be a solid, liquid or gas under certain conditions? We have to know about the forces that hold the material together.
In solids, the separate molecules or ions are held tightly in their positions by some type of force. They might vibrate a little bit in place, but they don't move around. For this reason, solids don't easily change shape. You might be able to bend or break them, but usually they don't change shape by themselves.
In liquids, the separate molecules or ions are held close together by some type of force, but they can move around while staying close together. For this reason, a liquid can change shape to fit whatever is holding it. But like solids, liquids have approximately constant volume. Even if the liquid flows into a new shape, the distance between the molecules doesn't change, so the total volume stays the same.
In a gas, usually the molecules bounce around as though there are no forces between them. (See Kinetic-Molecular Theory of Gases .) At very high pressure, when they are forced to be close together, we might start to notice that there are some forces between the molecules (because the pressure is less than we expect from the Ideal Gas Law) but usually they move around separately. Because they aren't really attracted to each other and have a lot of kinetic energy (at least at normal temperatures), they fill the whole space they have. So gases can take any shape, and also can change volume a lot.
The higher the temperature, the more kinetic energy the molecules or ions have.  So as we increase the temperature, we might see phase transitions from solid to liquid to gas. We can think about these transitions using equilibrium, like when we think about reaction and solubility equilibria.  Some of the molecules have bigger kinetic energy and some have smaller kinetic energy. If a molecule with big kinetic energy is on the surface of the liquid, it might fly off and enter the gas state. At the same time, other molecules in the gas state might bump the surface of the liquid, and if they don't have very much kinetic energy, they might stay there and join the liquid. This is a dynamic equilibrium: the molecules go back and forth between the 2 states. If we increase the temperature, the average kinetic energy increases, and that means the molecules are more likely to have enough kinetic energy to go into or stay in the gas phase. Liquid molecules will become gas molecules more often, and gas molecules will become liquid molecules less often. Then the equilibrium will move, so a bigger % of the total molecules are gas.
In many cases, there is a specific temperature above which all of a material goes from solid to liquid (a melting point) or from liquid to gas (a boiling point). What temperature that is depends on the strength and type of forces between the molecules and ions. If the forces are strong, then more kinetic energy is needed to make the molecules move around or separate and become gas, which means the melt point and boiling point are higher. If the forces between molecules are very weak, then the material may be a gas, and it may be hard to cool it enough to make a liquid.
In the next sections, we will talk about the forces between molecules that determine boiling points and melting points and other important properties. (Thus, they are called intermolecular forces , to separate them from the forces inside molecules that hold the molecules together.) These forces are often called Van der Waals forces after Johannes van der Waals, who wrote the equation for real gases . Van der Waals figured out that the reason gas pressures are often lower than we expect (at high pressure) is because there are attractions between the molecules. Actually Van der Waals was the son of a carpenter who wasn't allowed to enter university because he didn't have the right expensive primary education. But he took classes anyway, became a teacher, and eventually they changed the rules for university admission, so he got a doctorate, became famous, became a professor, and won the Nobel Prize.


9.2: Exercise
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The importance of getting regular exercise is probably nothing new to you. The health benefits are well known and established: Regular physical activity can produce long-term health benefits by reducing your risk of many health problems, such as heart disease, cancer, and diabetes, and it can also increase your chances of living longer, help you control your weight, and even help you sleep better.
As a busy college student, you may be thinking, I know this, but I don’t have time! I have classes and work and a full life!  Getting into an effective exercise routine now will not only make it easier to build healthy habits that you can take with you into your life after college, but it can actually help you be a more successful student, too. As you’ll see in the section on brain health, below, exercise is a powerful tool for improving one’s mental health and memory—both of which are especially important when you’re in school.
The good news is that most people can improve their health and quality of life through a modest increase in daily activity. You don’t have to join a gym, spend a lot of money, or even do the same activity every time—just going for a walk or choosing to take the stairs (instead of the elevator) can make a difference. The following video describes how much activity you need.
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Our bodies have an automatic “fight or flight” reaction when we perceive a threat. Fighting or running is physical exercise and the result is metabolizing our excessive stress hormones and bringing our bodies and minds back to a more relaxed state. Even though we do not suffer from the same threats cavemen and cavewomen incurred many years ago, we still perceive threats and we still suffer from stress. Exercise has many physical and mental benefits in addition to lowering stress.
Regular physical activity is one of the best things you can do to be healthy. According to the Centers for Disease Control and Prevention, its benefits include: controlling weight, reducing risk of cardiovascular disease, reducing risk for type two diabetes, reducing risk of cancer, strengthening bones and muscles, improving mental health and mood, increasing chances of living longer [1] . One assignment I give my students is to have them record all food and drinks consumed in one week, along with keeping records of how much exercise and how many hours of sleep take place. Some students know exactly what they are putting into their body and how they are treating it. But most are surprised at how little one or more of these important aspects are sufficient. Try it. Think of it as an opportunity to see what you actually put into your body, how much exercise and rest you give it.
I had not prepared for an astronomy final exam. The night before the exam, I stayed up all night cramming, and regurgitated information I had learned onto the test. Immediately after the exam, I ordered a large pepperoni pizza, ate the whole thing in one sitting, and then slept for 15 hours. Regardless of the exam result, this example represents what not to do. In the long run, human bodies cannot do things like that without consequence. With the lifestyle I live now, being married, parenting kids and working full-time, there is no way that I could have done that and been able to function well with my responsibilities the next day. In addition, it was not healthy for my body.
After a class I was speaking with a student about diet and nutrition and our conversation led to a discussion on the contrast of what some students give their children to eat versus what they eat themselves. It reminded me of the 5-2-1-0 graphic my children recognize at their pediatrician visits. It stands for (each day) 5 or more fruits or vegetables, 2 hours or less recreational screen time, 1 hour or more of physical activity, 0 sugary drinks (more water). Of course, it is designed for kids. But many students would do well for themselves to keep the 5-2-1-0 recommendation in mind [2] .
Our bodies are more prone to getting sick if they are not well taken care of. Getting sick in the middle of an academic term can have devastating effects on the academic performance.
A much better way to go (as I painfully learned) is to make a schedule, stick to it, prepare and review periodically, get adequate sleep, eat well, and be on an exercise plan.
For optimal concentration levels, work performance and test scores, proper nutrition and adequate sleep have a large effect. My personal opinion is that exercise helps clear the mind, which is so important in our current generations’ overwhelming bombardment of information around us.
Physical fitness is a state of well-being that gives you sufficient energy to perform daily physical activities without getting overly tired or winded. It also means being in good enough shape to handle unexpected emergencies involving physical demands—that is, if someone said, “Run for your life!” or you had to rush over and prevent a child from falling, you’d be able to do it.
There are many forms of exercise—dancing, rock climbing, walking, jogging, yoga, bike riding, you name it—that can help you become physically fit. The major types are described below.
Aerobic exercise increases your heart rate, works your muscles, and raises your breathing rate. For most people, it’s best to aim for a total of about thirty minutes a day, four or five days a week. If you haven’t been very active recently, you can start out with five or ten minutes a day and work up to more time each week. Or, split up your activity for the day: try a brisk ten-minute walk after each meal. If you are trying to lose weight, you may want to exercise more than thirty minutes a day. The following are some examples of aerobic exercise:
Strength training, done several times a week, helps build strong bones and muscles and makes everyday chores like carrying heavy backpacks (or grocery bags) easier. When you have more muscle mass, you burn more calories, even at rest. Here are some ways to do it:
Flexibility exercises, also called stretching, help keep your joints flexible and reduce your risk of injury during other activities. Gentle stretching for 5 to 10 minutes helps your body warm up and get ready for aerobic activities such as walking or swimming. Check to see if your college offers yoga, stretching, and/or pilates classes, and give one a try.
In addition to formal exercise, there are many opportunities to be active throughout the day. Being active helps burns calories. The more you move around, the more energy you will have. The following strategies can help you increase your activity level:
Exercise, even after age fifty, can add healthy, active years to one’s life. Studies continue to show that it’s never too late to start exercising and that even small improvements in physical fitness can significantly lower the risk of death. Simply walking regularly can prolong your life.
Moderately fit people—even if they smoke or have high blood pressure—have a lower mortality rate than the least fit. Resistance training is important because it’s the only form of exercise that can slow and even reverse the decline of muscle mass, bone density, and strength. Adding workouts that focus on speed and agility can be especially protective for older people. Flexibility exercises help reduce the stiffness and loss of balance that accompanies aging.
Diabetes, particularly type 2, is reaching epidemic proportions throughout the world as more and more cultures adopt Western-style diets (which tend to be high in sugar and fat). Aerobic exercise is proving to have significant and particular benefits for people with both type 1 and type 2 diabetes; it increases sensitivity to insulin, lowers blood pressure, improves cholesterol levels, and decreases body fat. In fact, studies show that people who engage in regular, moderate aerobic exercise (e.g., brisk walking, biking) lower their risk for diabetes even if they do not lose weight. Anyone on insulin or who has complications from diabetes should get advice from a physician before embarking on a workout program.
In addition to keeping your heart healthy, helping with weight loss, and helping you live longer, regular exercise can also improve your mood and help keep depression and anxiety at bay. The following video explains why and challenges you to give it a try:
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If you still aren’t persuaded, check out this slightly longer but excellent Tedx Talk, which describes how aerobic exercise can improve your cognitive functioning, memory, and creativity:
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