13.8.7: Higher Roots
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Learning Objectives
By the end of this section, you will be able to:
- Simplify expressions with higher roots
- Use the Product Property to simplify expressions with higher roots
- Use the Quotient Property to simplify expressions with higher roots
- Add and subtract higher roots
Note
-
Simplify: \(y^{5}y^{4}\).
If you missed this problem, review Example 6.2.7 . -
Simplify: \((n^2)^6\).
If you missed this problem, review Example 6.2.19 . -
Simplify: \(\frac{x^8}{x^3}\).
If you missed this problem, review Example 6.5.1 .
Simplify Expressions with Higher Roots
Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.
Let’s review some vocabulary first.
\[\begin{array}{cc} {}&{}\\ {\textbf{We write:}}&{\textbf{We say:}}\\ {n^2}&{\text{n squared}}\\ {n^3}&{\text{n cubed}}\\ {n^4}&{\text{n to the fourth}}\\ {n^5}&{\text{n to the fifth}}\\ \nonumber \end{array}\]
The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.
It will be helpful to have a table of the powers of the integers from −5to5. See Figure \(\PageIdnex{1}\).
Notice the signs in Figure \(\PageIndex{1}\). All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 below to help you see this.
Earlier in this chapter we defined the square root of a number.
If \(n^2=m\),then n is a square root of m.
And we have used the notation \(\sqrt{m}\) to denote the principal square root . So \(\sqrt{m} \ge 0\) always.
We will now extend the definition to higher roots.
Definition: N TH ROOT OF A NUMBER
If \(b^n=a\), then b is an n th root of a number a.
The principal n th root of a is written \(\sqrt[n]{a}=b\)
n is called the index of the radical.
We do not write the index for a square root. Just like we use the word ‘cubed’ for \(b^3\), we use the term ‘cube root’ for \(\sqrt[3]{a}\).
We refer to Figure \(\PageIndex{1}\) to help us find higher roots.
\[\begin{array}{cc} {4^3=64}&{\sqrt[3]{64}=4}\\ {3^4=81}&{\sqrt[4]{81}=3}\\ {(−2)^5=−32}&{\sqrt[5]{−32}=−2}\\ \nonumber \end{array}\]
Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.
Definition: PROPERTIES OF \(\sqrt[n]{a}\)
When n is an even number and
- \(a\ge 0\), then \(\sqrt[n]{a}\) is a real number
- \(a < 0\), then \(\sqrt[n]{a}\) is not a real number
When n is an odd number, \(\sqrt[n]{a}\) is a real number for all values of a.
Example \(\PageIndex{1}\)
Simplify:
- \(\sqrt[3]{8}\)
- \(\sqrt[4]{81}\)
- \(\sqrt[5]{32}\).
- Answer
-
1. \(\sqrt[3]{8}\) Since \((2)^3=8\). 2 2. \(\sqrt[4]{81}\) Since \((3)^4=81\). 3 3. \(\sqrt[5]{32}\) Since \((2)^5=32\). 2
Example \(\PageIndex{2}\)
Simplify:
- \(\sqrt[3]{27}\)
- \(\sqrt[4]{256}\)
- \(\sqrt[5]{243}\).
- Answer
-
- 3
- 4
- 3
Example \(\PageIndex{3}\)
Simplify:
- \(\sqrt[3]{1000}\)
- \(\sqrt[4]{16}\)
- \(\sqrt[5]{32}\).
- Answer
-
- 10
- 2
- 2
Example \(\PageIndex{4}\)
Simplify:
- \(\sqrt[3]{−64}\)
- \(\sqrt[4]{−16}\)
- \(\sqrt[5]{−243}\).
- Answer
-
1. \(\sqrt[3]{−64}\) Since \((−4)^3=−64\). −4 2. \(\sqrt[4]{−16}\) Think, \((?)^4=−16\).No real number raised to the fourth power is positive. Not a real number. 3. \(\sqrt[5]{−243}\) Since \((−3)^5=−243\). −3
Example \(\PageIndex{5}\)
Simplify:
- \(\sqrt[3]{−125}\)
- \(\sqrt[4]{−16}\)
- \(\sqrt[5]{−32}\).
- Answer
-
- −5
- not real
- −2
Example \(\PageIndex{6}\)
Simplify:
- \(\sqrt[3]{−216}\)
- \(\sqrt[4]{−81}\)
- \(\sqrt[5]{−1024}\).
- Answer
-
- −6
- not real
- −4
The odd root of a number can be either positive or negative. We have seen that \(\sqrt[3]{−64}=−4\).
But the even root of a non-negative number is always non-negative, because we take the principal n th root .
Suppose we start with a=−5.
\[\begin{array}{cc} {(−5)^4=625}&{\sqrt[4]{625}=5}\\ \nonumber \end{array}\]
How can we make sure the fourth root of −5 raised to the fourth power, \((−5)^4\) is 5? We will see in the following property.
Definition: SIMPLIFYING ODD AND EVEN ROOTS
For any integer \(n \ge 2\),
\[\begin{array}{cc} {\text{when n is odd}}&{\sqrt[n]{a^n}=a}\\ {\text{when n is even}}&{\sqrt[n]{a^n}=|a|}\\ \nonumber \end{array}\]
We must use the absolute value signs when we take an even root of an expression with a variable in the radical.
Example \(\PageIndex{7}\)
Simplify:
- \(\sqrt{x^2}\)
- \(\sqrt[3]{n^3}\)
- \(\sqrt[4]{p^4}\)
- \(\sqrt[5]{y^5}\).
- Answer
-
We use the absolute value to be sure to get the positive root.
1. \(\sqrt{x^2}\) Since \((x)^2=x^2\) and we want the positive root. |x| 2. \(\sqrt[3]{n^3}\) Since \((n)^3=n^3\). It is an odd root so there is no need for an absolute value sign. n 3. \(\sqrt[4]{p^4}\) Since \((p)^4=p^4\) and we want the positive root. |p| 4. \(\sqrt[5]{y^5}\) Since \((y)^5=y^5\). It is an odd root so there is no need for an absolute value sign. y
Example \(\PageIndex{8}\)
Simplify:
- \(\sqrt{b^2}\)
- \(\sqrt[3]{w^3}\)
- \(\sqrt[4]{m^4}\)
- \(\sqrt[5]{q^5}\).
- Answer
-
- |b|
- w
- |m|
- q
Example \(\PageIndex{9}\)
Simplify:
- \(\sqrt{y^2}\)
- \(\sqrt[3]{p^3}\)
- \(\sqrt[4]{z^4}\)
- \(\sqrt[5]{q^5}\)
- Answer
-
- |y|
- p
- |z|
- q
Example \(\PageIndex{10}\)
Simplify:
- \(\sqrt[3]{y^{18}}\)
- \(\sqrt[4]{z^8}\).
- Answer
-
1. \(\sqrt[3]{y^{18}}\) Since \((y^6)^3=y^18\). \(\sqrt[3]{(y^6)^3}\) \(y^6\) 2. \(\sqrt[4]{z^8}\) Since \((z^2)^4=z^8\). \(\sqrt[4]{(z^2)^4}\) Since \(z^2\) is positive, we do not need an absolute value sign. \(z^2\)
Example \(\PageIndex{11}\)
Simplify:
- \(\sqrt[4]{u^{12}}\)
- \(\sqrt[3]{v^{15}}\).
- Answer
-
- \(u^3\)
- \(v^5\)
Example \(\PageIndex{12}\)
Simplify:
- \(\sqrt[5]{c^{20}}\)
- \(\sqrt[6]{d^{24}}\).
- Answer
-
- \(c^4\)
- \(d^4\)
Example \(\PageIndex{13}\)
Simplify:
- \(\sqrt[3]{64p^6}\)
- \(\sqrt[4]{16q^{12}}\).
- Answer
-
1. \(\sqrt[3]{64p^6}\) Rewrite \(64p^6\) as \((4p^2)^3\). \(\sqrt[3]{(4p^2)^3}\) Take the cube root. \(4p^2\) 2. \(\sqrt[4]{16q^{12}}\) Rewrite the radicand as a fourth power. \(\sqrt[4]{(2q^3)^4}\) Take the fourth root. \(2|q^3|\)
Example \(\PageIndex{14}\)
Simplify:
- \(\sqrt[3]{27x^{27}}\)
- \(\sqrt[4]{81q^{28}}\).
- Answer
-
- \(3x^9\)
- \(3∣q^7∣\)
Example \(\PageIndex{15}\)
Simplify:
- \(\sqrt[3]{125p^9}\)
- \(\sqrt[5]{243q^{25}}\)
- Answer
-
- \(5p^3\)
- \(3q^5\)
Use the Product Property to Simplify Expressions with Higher Roots
We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An n th root is considered simplified if it has no factors of \(m^n\).
Definition: SIMPLIFIED N TH ROOT
\(\sqrt[n]{a}\) is considered simplified if a has no factors of \(m^n\).
We will generalize the Product Property of Square Roots to include any integer root \(n \ge 2\).
Definition: PRODUCT PROPERTY OF N TH ROOTs
\(\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b}\) and \(\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}\)
when \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers and for any integer \(n \ge 2\)
Example \(\PageIndex{16}\)
Simplify:
- \(\sqrt[3]{x^4}\)
- \(\sqrt[4]{x^7}\).
- Answer
-
1.
\(\sqrt[3]{x^4}\) Rewrite the radicand as a product using the largest perfect cube factor. \(\sqrt[3]{x^3·x}\) Rewrite the radical as the product of two radicals. \(\sqrt[3]{x^3}·\sqrt[3]{x}\) Simplify. \(x\sqrt[3]{x}\) 2. \(\sqrt[4]{x^7}\) Rewrite the radicand as a product using the greatest perfect fourth power factor. \(\sqrt[4]{x^4·x^3}\) Rewrite the radical as the product of two radicals. \(\sqrt[4]{x^4}·\sqrt[4]{x^3}\) Simplify. \(|x|\sqrt[4]{x^3}\)
Example \(\PageIndex{17}\)
Simplify:
- \(\sqrt[4]{y^6}\)
- \(\sqrt[3]{z^5}\).
- Answer
-
- \(|y∣\sqrt[4]{y^2}\)
- \(z\sqrt[3]{z^2}\)
Example \(\PageIndex{18}\)
Simplify:
- \(\sqrt[5]{p^8}\)
- \(\sqrt[6]{q^{13}}\).
- Answer
-
- \(p\sqrt[5]{p^3}\)
- \( q^2\sqrt[6]{q}\)
Example \(\PageIndex{19}\)
Simplify:
- \(\sqrt[3]{16}\)
- \(\sqrt[4]{243}\).
- Answer
-
1. \(\sqrt[3]{16}\) \(\sqrt[3]{2^4}\) Rewrite the radicand as a product using the largest perfect cube factor. \(\sqrt[3]{2^3·2}\) Rewrite the radical as the product of two radicals. \(\sqrt[3]{2^3}·\sqrt[3]{2}\) Simplify. \(2\sqrt[3]{2}\) 2. \(\sqrt[4]{243}\) \(\sqrt[4]{3^5}\) Rewrite the radicand as a product using the greatest perfect fourth power factor. \(\sqrt[4]{3^4·3}\) Rewrite the radical as the product of two radicals. \(\sqrt[4]{3^4}·\sqrt[4]{3}\) Simplify. \(3\sqrt[4]{3}\)
Example \(\PageIndex{20}\)
Simplify:
- \(\sqrt[3]{81}\)
- \(\sqrt[4]{64}\).
- Answer
-
- \(3\sqrt[3]{3}\)
- \(2\sqrt[4]{4}\)
Example \(\PageIndex{21}\)
Simplify:
- \(\sqrt[3]{625}\)
- \(\sqrt[4]{729}\).
- Answer
-
- \(5\sqrt[3]{5}\)
- \(3\sqrt[4]{9}\)
Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.
Example \(\PageIndex{22}\)
Simplify:
- \(\sqrt[3]{24x^7}\)
- \(\sqrt[4]{80y^{14}}\).
- Answer
-
1. \(\sqrt[3]{24x^7}\) Rewrite the radicand as a product using perfect cube factors. \(\sqrt[3]{2^{3}x^{6}·3x}\) Rewrite the radical as the product of two radicals. \(\sqrt[3]{2^{3}x^{6}}·\sqrt[3]{3x}\) Rewrite the first radicand as \((2x^2)^3\) \(\sqrt[3]{(2x^{2})^3}·\sqrt[3]{3x}\) Simplify. \(2x^2\sqrt[3]{3x}\) 2. \(\sqrt[4]{80y^{14}}\) Rewrite the radicand as a product using perfect fourth power factors. \(\sqrt[4]{2^{4}y^{12}·5y^2}\) Rewrite the radical as the product of two radicals. \(\sqrt[4]{2^{4}y^{12}}·\sqrt[4]{5y^2}\) Rewrite the first radicand as \((2y^3)^4\) \(\sqrt[4]{(2y^3)^4}·\sqrt[4]{5y^2}\) Simplify. \(2|y^3|\sqrt[4]{5y^2}\)
Example \(\PageIndex{23}\)
Simplify:
- \(\sqrt[3]{54p^[10}]\)
- \(\sqrt[4]{64q^{10}}\).
- Answer
-
- \(3p^3\sqrt[3]{2p}\)
- \(2q^2\sqrt[4]{4q^2}\)
Example \(\PageIndex{24}\)
Simplify:
- \(\sqrt[3]{128m^{11}}\)
- \(\sqrt[4]{162n^7}\).
- Answer
-
- \(4m^3\sqrt[3]{2m^2}\)
- \(3|n|\sqrt[4]{2n^3}\)
Example \(\PageIndex{25}\)
Simplify:
- \(\sqrt[3]{−27}\)
- \(\sqrt[4]{−16}\).
- Answer
-
1. \(\sqrt[3]{−27}\) Rewrite the radicand as a product using perfect cube factors. \(\sqrt[3]{(−3)^3}\) Take the cube root. −3 2. \(\sqrt[4]{−16}\) There is no real number n where \(n^4=−16\). Not a real number.
Example \(\PageIndex{26}\)
Simplify:
- \(\sqrt[3]{−108}\)
- \(\sqrt[4]{−48}\).
- Answer
-
- \(−3\sqrt[3]{4}\)
- not real
Example \(\PageIndex{27}\)
Simplify:
- \(\sqrt[3]{−625}\)
- \(\sqrt[4]{−324}\).
- Answer
-
- \(−5\sqrt[3]{5}\)
- not real
Use the Quotient Property to Simplify Expressions with Higher Roots
We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.
Example \(\PageIndex{28}\)
Simplify:
- \(\sqrt[3]{\frac{a^8}{a^5}}\)
- \(\sqrt[4]{\frac{a^{10}}{a^2}}\).
- Answer
-
1.
\(\sqrt[3]{\frac{a^8}{a^5}}\) Simplify the fraction under the radical first. \(\sqrt[3]{a^3}\) Simplify. a 2. \(\sqrt[4]{\frac{a^{10}}{a^2}}\) Simplify the fraction under the radical first. \(\sqrt[4]{a^8}\) Rewrite the radicand using perfect fourth power factors. \(\sqrt[4]{(a^2)^4}\) Simplify. \(a^2\)
Example \(\PageIndex{29}\)
Simplify:
- \(\sqrt[4]{\frac{x^7}{x^3}}\)
- \(\sqrt[4]{\frac{y^{17}}{y^5}}\).
- Answer
-
- |x|
- \(y^3\)
Example \(\PageIndex{30}\)
Simplify:
- \(\sqrt[3]{\frac{m^{13}}{m^7}}\)
- \(\sqrt[5]{\frac{n^{12}}{n^2}}\).
- Answer
-
- \(m^2\)
- \(n^2\)
Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.
Definition: QUOTIENT PROPERTY OF N TH ROOTS
\(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) and \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
when \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers, \(b \ne 0\), and for any integer \(n \ge 2\)
Exercise \(\PageIndex{31}\)
Simplify:
- \(\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}\)
- \(\frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}\)
- Answer
-
1. \(\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}\) Neither radicand is a perfect cube, so use the Quotient Property to write as one radical. \(\sqrt[3]{\frac{−108}{2}}\) Simplify the fraction under the radical. \(\sqrt[3]{−54}\) Rewrite the radicand as a product using perfect cube factors. \(\sqrt[3]{(−3)^3·2}\) Rewrite the radical as the product of two radicals. \(\sqrt[3]{(−3)^3}·\sqrt[3]{2}\) Simplify. \(−3\sqrt[3]{2}\) 2. \(\frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}\) Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical \(\sqrt[4]{\frac{96x^7}{3x^2}}\) Simplify the fraction under the radical. \(\sqrt[4]{32x^5}\) Rewrite the radicand as a product using perfect fourth power factors. \(\sqrt[4]{2^{4}x^4·2x}\) Rewrite the radical as the product of two radicals. \(\sqrt[4]{(2x)^4}·\sqrt[4]{2x}\) Simplify. \(2|x|\sqrt[4]{2x}\)
Example \(\PageIndex{32}\)
Simplify:
- \(\frac{\sqrt[3]{−532}}{\sqrt[3]{2}}\)
- \(\frac{\sqrt[4]{486m^{11}}}{\sqrt[4]{3m^5}}\)
- Answer
-
- not real
- \(3|m|\sqrt[4]{2m^2}\)
Example \(\PageIndex{33}\)
Simplify:
- \(\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}\)
- \(\frac{\sqrt[4]{324n^7}}{\sqrt[4]{2n^3}}\).
- Answer
-
- −4
- \(3|n|\sqrt[4]{2}\)
If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.
Example \(\PageIndex{34}\)
Simplify:
- \(\sqrt[3]{\frac{24x^7}{y^3}}\)
- \(\sqrt[4]{\frac{48x^{10}}{y^8}}\).
- Answer
-
1. \(\sqrt[3]{\frac{24x^7}{y^3}}\) The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals. \(\frac{\sqrt[3]{24x^7}}{\sqrt[3]{y^3}}\) Rewrite each radicand as a product using perfect cube factors. \(\frac{\sqrt[3]{8x^6·3x}}{\sqrt[3]{y^3}}\) Rewrite the numerator as the product of two radicals. \(\frac{\sqrt[3]{(2x^2)^3}·\sqrt[3]{3x}}{\sqrt[3]{y^3}}\) Simplify. \(\frac{2x^2\sqrt[3]{3x}}{y}\) 2. \(\sqrt[4]{\frac{48x^{10}}{y^8}}\) The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals. \(\frac{\sqrt[4]{48x^{10}}}{\sqrt[4]{y^8}}\) Rewrite each radicand as a product using perfect cube factors. \(\frac{\sqrt[4]{16x^8·3x^2}}{\sqrt[4]{y^8}}\) Rewrite the numerator as the product of two radicals. \(\frac{\sqrt[4]{(2x^2)^4}·\sqrt[4]{3x^2}}{\sqrt[4]{(y^2)^4}}\) Simplify. \(\frac{2x^2\sqrt[4]{3x^2}}{y^2}\)
Example \(\PageIndex{35}\)
Simplify:
- \(\sqrt[3]{\frac{108c^{10}}{d^6}}\)
- \(\sqrt[4]{\frac{80x^{10}}{y^5}}\).
- Answer
-
- \(\frac{3c^3\sqrt[3]{4c}}{d^2}\)
- \(\frac{x^2}{∣y∣}\sqrt[4]{\frac{80x^2}{y}}\)
Example \(\PageIndex{36}\)
Simplify:
- \(\sqrt[3]{\frac{40r^3}{s}}\)
- \(\sqrt[4]{\frac{162m^{14}}{n^{12}}}\)
- Answer
-
- \(r\sqrt[3]{\frac{40}{s}}\)
- \(\frac{3m^3\sqrt[4]{2m^2}}{∣n^3∣}\)
Add and Subtract Higher Roots
We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals .
Definition: LIKE RADICALS
Radicals with the same index and same radicand are called like radicals .
Like radicals have the same index and the same radicand.
- \(9\sqrt[4]{42x}\) and \(−2\sqrt[4]{42x}\) are like radicals.
- \(5\sqrt[3]{125x}\) and \(6\sqrt[3]{125y}\) are not like radicals. The radicands are different.
- \(2\sqrt[5]{1000q}\) and \(−4\sqrt[4]{1000q}\) are not like radicals. The indices are different.
We add and subtract like radicals in the same way we add and subtract like terms. We can add \(9\sqrt[4]{42x}+(−2\sqrt[4]{42x})\) and the result is \(7\sqrt[4]{42x}\).
Example \(\PageIndex{37}\)
Simplify:
- \(\sqrt[3]{4x}+\sqrt[3]{4x}\)
- \(4\sqrt[4]{8}−2\sqrt[4]{8}\)
- Answer
-
1. \(\sqrt[3]{4x}+\sqrt[3]{4x}\) The radicals are like, so we add the coefficients \(2\sqrt[3]{4x}\) 2. \(4\sqrt[4]{8}−2\sqrt[4]{8}\) The radicals are like, so we subtract the coefficients. \(2\sqrt[4]{8}\)
Example \(\PageIndex{38}\)
Simplify:
- \(\sqrt[5]{3x}+\sqrt[5]{3x}\)
- \(3\sqrt[3]{9}−\sqrt[3]{9}\)
- Answer
-
- \(2\sqrt[5]{3x}\)
- \(2\sqrt[3]{9}\)
Example \(\PageIndex{39}\)
Simplify:
- \(\sqrt[4]{10y}+\sqrt[4]{10y}\)
- \(5\sqrt[6]{32}−3\sqrt[6]{32}\).
- Answer
-
- \(2\sqrt[4]{10y}\)
- \(2\sqrt[6]{32}\)
When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.
Example \(\PageIndex{40}\)
Simplify:
- \(\sqrt[3]{54}−\sqrt[3]{16}\)
- \(\sqrt[4]{48}+\sqrt[4]{243}\).
- Answer
-
1. \(\sqrt[3]{54}−\sqrt[3]{16}\) Rewrite each radicand using perfect cube factors. \(\sqrt[3]{27}·\sqrt[3]{2}−\sqrt[3]{8}·\sqrt[3]{2}\) Rewrite the perfect cubes. \(\sqrt[3]{(3)^3}·\sqrt[3]{2}−\sqrt[3]{(2)^3}·\sqrt[3]{2}\) Simplify the radicals where possible. \(3\sqrt[3]{2}−2\sqrt[3]{2}\) Combine like radicals. \(\sqrt[3]{2}\) 2. \(\sqrt[4]{48}+\sqrt[4]{243}\) Rewrite using perfect fourth power factors. \(\sqrt[4]{16}·\sqrt[4]{3}+\sqrt[4]{81}·\sqrt[4]{3}\) Rewrite each radicand as a product using perfect cube factors. \(\sqrt[4]{(2)^4}·\sqrt[4]{3}+\sqrt[4]{(3)^4}·\sqrt[4]{3}\) Rewrite the numerator as the product of two radicals. \(2\sqrt[4]{3}+3\sqrt[4]{3}\) Simplify. \(5\sqrt[4]{3}\)
Example \(\PageIndex{41}\)
Simplify:
- \(\sqrt[3]{192}−\sqrt[3]{81}\)
- \(\sqrt[4]{32}+\sqrt[4]{512}\).
- Answer
-
- \(\sqrt[3]{3}\)
- \(6\sqrt[4]{2}\)
Example \(\PageIndex{42}\)
Simplify:
- \(\sqrt[3]{108}−\sqrt[3]{250}\)
- \(\sqrt[5]{64}+\sqrt[5]{486}\).
- Answer
-
- \(−\sqrt[3]{2}\)
- \(5\sqrt[5]{2}\)
Example \(\PageIndex{43}\)
Simplify:
- \(\sqrt[3]{24x^4}−\sqrt[3]{−81x^7}\)
- \(\sqrt[4]{162y^9}+\sqrt[4]{512y^5}\).
- Answer
-
1. \(\sqrt[3]{24x^4}−\sqrt[3]{−81x^7}\) Rewrite each radicand using perfect cube factors. \(\sqrt[3]{8x^3}·\sqrt[3]{3x}−\sqrt[3]{−27x^6}·\sqrt[3]{3x}\) Rewrite the perfect cubes. \(\sqrt[3]{(2x)^3}·\sqrt[3]{3x}−\sqrt[3]{(−3x^2)^3}·\sqrt[3]{3x}\) Simplify the radicals where possible. \(2x\sqrt[3]{3x}−(−3x^2\sqrt[3]{3x})\) 2. \(\sqrt[4]{162y^9}+\sqrt[4]{516y^5}\) Rewrite using perfect fourth power factors. \(\sqrt[4]{81y^8}·\sqrt[4]{2y}+\sqrt[4]{256y^4}·\sqrt[4]{2y}\) Rewrite each radicand as a product using perfect cube factors. \(\sqrt[4]{(3y^2)^4}·\sqrt[4]{2y}+\sqrt[4]{(4y)^4}·\sqrt[4]{2y}\) Rewrite the numerator as the product of two radicals. \(3y^2\sqrt[4]{2y}+4|y|\sqrt[4]{2y}\)
Example \(\PageIndex{44}\)
Simplify:
- \(\sqrt[3]{32y^5}−\sqrt[3]{−108y^8}\)
- \(\sqrt[4]{243r^{11}}+\sqrt[4]{768r^{10}}\).
- Answer
-
- \(2y\sqrt[3]{4y^2}+3y^2\sqrt[3]{4y^2}\)
- \(3r^2\sqrt[4]{3r^3}+4r^2\sqrt[4]{3r^2}\)
Example \(\PageIndex{45}\)
Simplify:
- \(\sqrt[3]{40z^7}−\sqrt[3]{−135z^4}\)
- \(\sqrt[4]{80s^{13}}+\sqrt[4]{1280s^6}\).
- Answer
-
- \(2z^2\sqrt[3]{5z}+3z^5\sqrt[3]{5z}\)
- \(2∣s^3∣\sqrt[4]{5s}+4|s|\sqrt[4]{5s}\)
- Simplifying Higher Roots
- Add/Subtract Roots with Higher Indices
Key Concepts
- Properties of
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\(\sqrt[n]{a}\) when n is an even number and
- \(a \ge 0\), then \(\sqrt[n]{a}\) is a real number
- \(a < 0\), then \(\sqrt[n]{a}\) is not a real number
- When n is an odd number, \(\sqrt[n]{a}\) is a real number for all values of a .
- For any integer \(n \ge 2\), when n is odd \(\sqrt[n]{a^n}=a\)
- For any integer \(n \ge 2\), when n is even \(\sqrt[n]{a^n}=|a|\)
- \(\sqrt[n]{a}\) is considered simplified if a has no factors of \(m^n\).
- \(\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b}\) and \(\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}\)
- \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) and \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
- To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.
Glossary
- n th root of a number
- If \(b^n=a\), then b is an n th root of a.
- principal n th root
- The principal n th root of a is written \(\sqrt[n]{a}\).
- index
- \(\sqrt[n]{a}\) n is called the index of the radical.
- like radicals
- Radicals with the same index and same radicand are called like radicals.


5.19: V Glossary
Dimensional analysis: Dimensional analysis is a problem-solving technique where measurements are converted to a different (but equivalent) unit of measure by multiplying with a fractional form of 1 to obtain a desired unit of administration.
Drop factor: The number of drops in one mL of solution when fluids or medications are administered using gravity IV tubing.
Equivalency: Two values or quantities that are the same amount. For example, one cup is equivalent to eight ounces.
Macrodrip tubing: Gravity IV tubing with drop factors of 10, 15, or 20 drops per milliliter that are typically used to deliver general IV solutions for adults.
Medication cup: A small plastic or paper cup used to dispense oral medications. Some plastic medication cups have calibration marks for measuring medication amounts.
Microdrip tubing: Gravity IV tubing with a drop factor of 60 drops per milliliter.
Military time: A method of measuring the time based on the full 24 hours of the day rather than two groups of 12 hours indicated by AM and PM.
Oral syringe: A specific type of syringe used to measure and/or administer medications via the oral route.
Reconstitution: The process of adding a liquid diluent to a dry ingredient to make a liquid in a specific concentration.
Syringe: A medical device used to administer parenteral medication into tissue or into the bloodstream. Syringes can also be used to withdraw blood or fluid.


1.2: Deterrence
It has been a popular notion throughout the ages that fear of punishment can reduce or eliminate undesirable behavior. This notion has always been popular among criminal justice thinkers. These ideas have been formalized in several different ways. The Utilitarian philosopher Jeremy Bentham is credited with articulating the three elements that must be present if deterrence is to work: The punishment must be administered with celerity, certainty, and appropriate severity. These elements are applied under a type rational choice theory . Rational choice theory is the simple idea that people think about committing a crime before they do it. If the rewards of the crime outweigh the punishment, then they do the prohibited act. If the punishment is seen as outweighing the rewards, then they do not do it. Sometimes criminologists borrow the phrase cost-benefit analysis from economists to describe this sort of decision-making process.
When evaluating whether deterrence works or not, it is important to differentiate between general deterrence and specific deterrence. General deterrence is the idea that every person punished by the law serves as an example to others contemplating the same unlawful act. Specific deterrence is the idea that the individuals punished by the law will not commit their crimes again because they "learned a lesson."
Critics of deterrence theory point to high recidivism rates as proof that the theory does not work. Recidivism means a relapse into crime. In other words, those who are punished by the criminal justice system tend to reoffend at a very high rate. Some critics also argue that rational choice theory does not work. They argue that such things as crimes of passion and crimes committed by those under the influence of drugs and alcohol are not the product of a rational cost-benefit analysis.
As unpopular as rational choice theories may be with particular schools of modern academic criminology, they are critically important to understanding how the criminal justice system works. This is because nearly the entire criminal justice system is based on rational choice theory . The idea that people commit crimes because they decide to do so is the very foundation of criminal law in the United States. In fact, the intent element must be proven beyond a reasonable doubt in almost every felony known to American criminal law before a conviction can be secured. Without a culpable mental state , there is no crime (with very few exceptions).


8.2: Three People on the Margins
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The first person: In 1761 a six-year-old girl was captured from West Africa, given the name Phillis Wheatley, and sold into slavery in the City of Boston in Colonial America. By the time she was 17, Phillis had taught herself to read and write and had developed a special love and talent for poetry. Her owner was a wealthy businessman and sought to improve his reputation by publishing an anthology of her poems. Unfortunately he encountered stiff resistance from publishers because few people at that time believed Africans to be capable of the thought and imagination needed to write poetry. People who heard of her poetry were skeptical and inclined to think that it was faked. Eventually, to save his own reputation, the owner assembled a tribunal of 18 prominent judges— including the governor of Massachusetts and John Hancock, one of the signers of the Declaration of Independence— to assess the young woman's mental capacity. After cross-examining her, the judges finally decided that Ms Wheatley was, after all, capable of writing poetry (Robinson, 1982).
The second person: A century later, a child named Helen Keller lost her sight and hearing as a result of illness during infancy. In spite of this misfortune, though, Helen devised a language of gestural signs for communicating with a tutor, and was soon also using Braille to study both French and Latin. At ten, she wrote and published a short story. Yet like Ms Wheatley, Ms Keller also faced substantial, chronic skepticism about her capacities. Prominent educators accused her of plagiarizing others' writings and merely "parroting" others' ideas without understanding them (Keller, 1954; Bogdan, 2006). Eventually, as with Wheatley, a panel was assembled— though this time the members were professional experts about disabilities— to determine whether Ms Keller was in fact capable of writing what she published. The panel decided that she was indeed capable, though only by a slim margin (five judges vs four).
The third person: In 1978, Sue Rubin was born with a disability that limited her speech to disordered bursts of sound and occasionally echoing phrases of other people.  With support and encouragement from her mother and others, however, Sue eventually learned to type on a keyboard without assistance. She learned to communicate effectively when she was about 13 and was able to go to school. Since then she has made many presentations about autism at conferences and recently co-edited a book about autism, titled Autism: The Myth of the Person Alone (Bogdan, et al., 2005).
One of these individuals experienced racial discrimination and the other two experienced physical disabilities, but notice something important: that all three were defined by society as disabled intellectually. Initially, their with special educational needs achievements were dismissed because of widespread assumptions— whether about race or disability— of their inherent incompetence.
Since the time of Phillis Wheatley, North American society has eliminated slavery and made some progress at reducing certain forms of racism, though much remains to be done. In 1954, for example, the United States Supreme Court ruled that public schools could not be segregated by race, and in doing so recognized, at least legally, the intellectual competence of African-Americans as well as the moral obligation of society to provide all citizens with the best possible education. It has taken longer to recognize legally the rights and competence of persons with disabilities, but events and trends beginning in the 1970s have begun to make it happen. This chapter begins by explaining some of these and how they have altered the work of teachers.


13.3: Physical Changes in Adolescence
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Adolescence begins with the onset of puberty , a developmental period in which hormonal changes cause rapid physical alterations in the body, culminating in sexual maturity. Although the timing varies to some degree across cultures, the average age range for reaching puberty is between 9 and 14 years for girls and between 10 and 17 years for boys (Marshall & Tanner, 1986). This period of physical development of the adolescent age 9-13 is divided into two phases. 9
The first phase of puberty begins when the pituitary gland begins to stimulate the production of the male sex hormone testosterone in boys and the female sex hormones estrogen and progesterone in girls.  It also involves height increases from 20 to 25 percent. Puberty is second to the prenatal period in terms of rapid growth as the long bones stretch to their final, adult size.
Secondary sexual characteristics are visible physical changes not directly linked to reproduction but signal sexual maturity. The growth spurt for girls usually occurs earlier than that for boys, with some boys continuing to grow into their 20s. For males this includes broader shoulders, an enlarged Adam’s apple, and a lower voice as the larynx grows. Boys typically begin to grow facial hair between ages 14 and 16, which becomes coarser and darker, and hair growth occurs in the pubic area, under the arms, and on the face.
For females the enlargement of breasts is usually the first sign of puberty and, on average, occurs between ages 10 and 12 (Marshall & Tanner, 1986). Girl’s hips broaden and pubic and underarm hair develops and becomes darker and coarser. Both boys and girls experience a rapid growth spurt during this stage. Males and females may begin shaving during this time period as well as showing signs of acne on their faces and bodies.
Acne is an unpleasant consequence of the hormonal changes in puberty. Acne is defined as pimples on the skin due to overactive sebaceous (oil-producing) glands (Dolgin, 2011). These glands develop at a greater speed than the skin ducts that discharge the oil. Consequently, the ducts can become blocked with dead skin and acne will develop. According to the University of California at Los Angeles Medical Center (2000), approximately 85% of adolescents develop acne, and boys develop acne more than girls because of greater levels of testosterone in their systems (Dolgin, 2011). Hormones that are also responsible for sexual development can also wreak havoc on the teenage skin.12
A major milestone in puberty for girls is menarche , the first menstrual period, typically experienced at around 12 or 13 years of age (Anderson, Dannal, & Must, 2003). The age of menarche varies substantially and is determined by genetics, as well as by diet and lifestyle, since a certain amount of body fat is needed to attain menarche. Girls who are very slim, who engage in strenuous athletic activities, or who are malnourished may begin to menstruate later. Even after menstruation begins, girls whose level of body fat drops below the critical level may stop having their periods. The sequence of events for puberty is more predictable than the age at which they occur. Some girls may begin to grow pubic hair at age 10 but not attain menarche until age 15.13
Male Anatomy
Males have both internal and external genitalia that are responsible for procreation and sexual intercourse.   The semen and sperm, as a result of sexual intercourse, can fertilize an ovum in the female's body; the fertilized ovum (zygote) develops into a fetus, which is later born as a child.
Female Anatomy
Female external genitalia is collectively known as the vulva, which includes the mons veneris, labia majora, labia minora, clitoris, vaginal opening, and urethral opening. Female internal reproductive organs consist of the vagina, uterus, fallopian tubes, and ovaries. The uterus hosts the developing fetus, produces vaginal and uterine secretions, and passes the male's sperm through to the fallopian tubes while the ovaries release the eggs. A female is born with all her eggs already produced. The vagina is attached to the uterus through the cervix, while the uterus is attached to the ovaries via the fallopian tubes. Females have a monthly reproductive cycle; at certain intervals the ovaries release an egg, which passes through the fallopian tube into the uterus. If, in this transit, it meets with sperm, the sperm might penetrate and merge with the egg, fertilizing it. If not fertilized, the egg and the tissue that was lining the uterus is flushed out of the system through menstruation (around every 28 days).
Effects of Puberty on Development
The age of puberty is getting younger for children throughout the world. A century ago the average age of a girl’s first period in the United States and Europe was 16, while today it is around 13. Because there is no clear marker of puberty for boys, it is harder to determine if boys are maturing earlier, too. In addition to better nutrition, less positive reasons associated with early puberty for girls include increased stress, obesity, and endocrine disrupting.
Because rates of physical development vary so widely among teenagers, puberty can be a source of pride or embarrassment. Girls and boys who develop more slowly than their peers may feel self-conscious about their lack of physical development; some research has found that negative feelings are particularly a problem for late maturing boys, who are at a higher risk for depression and conflict with parents (Graber et al., 1997) and more likely to be bullied (Pollack & Shuster, 2000). Additionally, problems are more likely to occur when the child is among the first in his or her peer group to develop. Because the preadolescent time is one of not wanting to appear different, early developing children stand out among their peer group and gravitate toward those who are older (Weir, 2016).
Early maturing boys tend to be physically stronger, taller, and more athletic than their later maturing peers; this can contribute to differences in popularity among peers, which can in turn influence the teenager’s confidence. Some studies show that boys who mature earlier tend to be more popular and independent but are also at a greater risk for substance abuse and early sexual activity (Flannery, Rowe, & Gulley, 1993; Kaltiala-Heino, Rimpela, Rissanen, & Rantanen, 2001).
Early maturing girls may face increased teasing and sexual harassment related to their developing bodies, which can contribute to self-consciousness and place them at a higher risk for anxiety, depression, substance abuse, and eating disorders (Ge, Conger, & Elder, 2001; Graber, Lewinsohn, Seeley, & Brooks-Gunn, 1997; Striegel-Moore & Cachelin, 1999).16
The Brain and Sex
The brain is the structure that translates the nerve impulses from the skin into pleasurable sensations. It controls nerves and muscles used during sexual behavior.  The cerebral cortex, which is the outer layer of the brain that allows for thinking and reasoning, is believed to be the origin of sexual thoughts and fantasies. Beneath the cortex is the limbic system, which consists of the amygdala, hippocampus, cingulate gyrus, and septal area. These structures are where emotions and feelings are believed to originate, and are important for sexual behavior.
 This is the small area at the base of the brain consisting of several groups of nerve-cell bodies that receives input from the limbic system. Studies with lab animals have shown that destruction of certain areas of the hypothalamus causes complete elimination of sexual behavior. One of the reasons for the importance of the hypothalamus is that it controls the pituitary gland, which secretes hormones that control the other glands of the body.
Hormones
Several important sexual hormones are secreted by the pituitary gland. Oxytocin, also known as the hormone of love, is released during sexual intercourse when an orgasm is achieved. Oxytocin is also released in females when they give birth or are breast-feeding; it is believed that oxytocin is involved with maintaining close relationships. Both prolactin and oxytocin stimulate milk production in females. Follicle-stimulating hormone (FSH) is responsible for ovulation in females by triggering egg maturity; it also stimulates sperm production in males. Luteinizing hormone (LH) triggers the release of a mature egg in females during the process of ovulation.
In males, testosterone appears to be a major contributing factor to sexual motivation. Vasopressin is involved in the male arousal phase, and the increase of vasopressin during erectile response may be directly associated with increased motivation to engage in sexual behavior.
The relationship between hormones and female sexual motivation is not as well understood, largely due to the overemphasis on male sexuality in Western research. Estrogen and progesterone typically regulate motivation to engage in sexual behavior for females, with estrogen increasing motivation and progesterone decreasing it. The levels of these hormones rise and fall throughout a woman's menstrual cycle. Research suggests that testosterone, oxytocin, and vasopressin are also implicated in female sexual motivation in similar ways as they are in males, but more research is needed to understand these relationships.
Sexuality will be discussed in Chapter 15, Adolescence Social Emotional Development. The following section will look at the reasons and the consequences of teenage pregnancy, forms of birth control, and sexually transmitted diseases.
Adolescent Pregnancy, Birth Control Methods, and Sexually Transmitted Infections
By the end of high school, more than half of boys and girls report having experienced sexual intercourse at least once, though it is hard to be certain of the proportion because of the sensitivity and privacy of the information. (Center for Disease Control, 2004; Rosenbaum, 2006).
Teen Pregnancy
Although adolescent pregnancy rates have declined since 1991, teenage birth rates in the United States are higher than most industrialized countries. In 2014, females aged 15–19 years experienced a birth rate of 24.2 per 1,000 women. This is a drop of 9% from 2013. Birth rates fell 11% for those aged 15–17 years and 7% for 18–19 year-olds. It appears that adolescents seem to be less sexually active than in previous years, and those who are sexually active seem to be using birth control (CDC, 2016).
Risk Factors for Adolescent Pregnancy
Miller, Benson, and Galbraith (2001) found that parent/child closeness, parental supervision, and parents' values against teen intercourse (or unprotected intercourse) decreased the risk of adolescent pregnancy. In contrast, residing in disorganized/dangerous neighborhoods, living in a lower SES family, living with a single parent, having older sexually active siblings or pregnant/parenting teenage sisters, early puberty, and being a victim of sexual abuse place adolescents at an increased risk of adolescent pregnancy.
Consequences of Adolescent Pregnancy
After a child is born life can be difficult for a teenage mother. Only 40% of teenagers who have children before age 18 graduate from high school. Without a high school degree, her job prospects are limited and economic independence is difficult. Teen mothers are more likely to live in poverty and more than 75% of all unmarried teen mothers receive public assistance within 5 years of the birth of their first child. Approximately, 64% of children born to an unmarried teenage high-school dropout live in poverty.  It’s important to mention that some STIs can also be contracted by sharing intravenous drug needles with an infected person, through childbirth, or breastfeeding. Common STIs include:
- chlamydia;
- herpes (HSV-1 and HSV-2);
- human papillomavirus (HPV);
- gonorrhea;
- syphilis;
- trichomoniasis;
- HIV (human immunodeficiency virus) and AIDS (acquired immunodeficiency syndrome).
According to the Centers for Disease Control and Prevention (CDC) (2014), there was an increase in the three most common types of STDs in 2014. Those most affected by STDS include younger, gay/bisexual males, and females. The most effective way to prevent transmission of STIs is to practice abstinence , (not participating in sexual intercourse), safe sex, and to avoid direct contact of skin or fluids which can lead to transfer with an infected partner. Proper use of safe-sex supplies (such as male condoms, female condoms, gloves, or dental dams) reduces contact and risk and can be effective in limiting exposure; however, some disease transmission may occur even with these barriers.23
Practicing safe sex is important to one’s physical health. In the following section we’ll look at elements of adolescent health, including sleep, diet, and exercise.
Contraceptive Methods and Protection from Sexually Transmitted Infection
There are many methods of contraception that sexually active adolescents can use to reduce the chances of pregnancy.
| Method | Description | Failure Rate |
|---|---|---|
| Intrauterine Contraception (IUD) | An IUD is a small device that is shaped in the form of a “T” placed inside the uterus | 0.1-0.8% |
| Implant |
A single, thin rod that is inserted under the skin of a woman’s upper arm. |
0.01% |
| Injection |
Injections or shots of hormones to prevent pregnancy are given in the buttocks or arm every three months. |
4% |
| Oral contraceptives | Also called “the pill,” contain the hormones to prevent pregnancy. A pill is taken at the same time each day. | 7% |
| Patch | This skin patch is worn on the lower abdomen, buttocks, or upper body and releases hormones to prevent pregnancy into the bloodstream. A new patch once a week for three weeks and then left off for a week. | 7% |
| Hormonal vaginal contraceptive ring | The ring is placed in the vagina and releases the hormones to prevent pregnancy. It is worn for three weeks. A week after it is removed a new ring is placed. | 7% |
| Spermicide | These kill sperm and come in several forms—foam, gel, cream, film, suppository, or tablet. They are placed in the vagina before intercourse. | 21% |
| Diaphragm or cervical cap | A cup that is placed inside the vagina to cover the cervix to block sperm. It is inserted with spermicide before sexual intercourse. | 17% |
| Sponge |
This contains spermicide and is placed in the vagina where it fits over the cervix. |
14-27% |
| Male condom | Worn (single use) by the man over the penis to keep sperm from getting into a woman’s body. | 13% |
| Female condom |
Worn (single use) by the woman inside the vagina to keep sperm from getting into a woman’s body. |
21% |
| Natural Family Planning | During a regular menstrual cycle, fertile days can be predicted. Sexual intercourse can be avoided on those days. | 2-23% |
| Copper IUD | Can be inserted up to 5 days after sexual intercourse | <1%25 |
| Emergency contraceptive pills | Can be taken up to 5 days after sexual intercourse and may be available over-the-counter. | 1-10%26 |
In choosing a birth control method, dual protection from the simultaneous risk for HIV and other STIs also should be considered. Although hormonal contraceptives and IUDs are highly effective at preventing pregnancy, they do not protect against STIs, including HIV. Consistent and correct use of the male latex condom reduces the risk for HIV infection and other STIs, including chlamydial infection, gonococcal infection, and trichomoniasis.
Contributors and Attributions
9. Adolescence: Developing Independence and Identity by Charles Stangor is licensed under CC BY-NC-SA 3.0
12. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
13. Beginning Psychology - Growing and Developing by Charles Stangor is licensed under CC BY-NC-SA 3.0
16. Adolescence Physical Growth by Boundless.com is licensed under CC BY-SA 4.0 ; Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
21. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
23. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
25. How effective is emergency contraception? (2016). Retrieved from https://www.nhs.uk/conditions/contraception/how-effective-emergency-contraception/
26. David G. Weismiller M.D., Sc.M (2004). Emergency Contraception. Retrieved from www.aafp.org/afp/2004/0815/p707.html


16.3: Identifying Trees
The following are equivalent for a graph \(G = (V,E)\) with \(\vert V \vert = n\) vertices.
- Graph \(G\) is a tree.
- Graph \(G\) is acyclic but the addition of any new edge would create a cycle.
- Graph \(G\) contains no loops and contains exactly one path between each pair of distinct vertices.
- Graph \(G\) is connected but every edge of \(G\) is a bridge.
- Graph \(G\) is connected and has exactly \(\vert E \vert = n-1\) edges.
- Graph \(G\) is acyclic and has exactly \(\vert E \vert = n-1\) edges.
- Proof of the equivalence of Statements 1–4.
-
Statement 1 implies Statement 2.
Suppose \(G\) is a tree. By definition, it is acyclic. Furthermore, suppose we add an edge between vertices \(v,v'\text{.}\) Since trees are connected, there was already a path from \(v\) to \(v'\) in \(G\text{.}\) Traversing the new edge from \(v'\) back to \(v\) closes that path to a cycle.
Statement 2 implies Statement 3.
Considering the contrapositive, we will assume that Statement 3 is false, and prove that this implies that Statement 2 must also be false.
For Statement 3 to be false, one of the following must be true.
- Loops exist in \(G\text{.}\)
- Some pair of distinct vertices in \(G\) is not connected.
- Some pair of distinct vertices in \(G\) is connected by more than one path.
In the first case, \(G\) would not be acyclic, as a loop is the most basic form of cycle. In the second case, adding an edge between these two vertices that were previously unconnected by a path would not create a cycle, as the rest of that cycle other than the new edge would have been a path between the two vertices. And in the third case, if a pair of vertices is connected by more than one path then the parts of two such paths that are different could be concatenated (one forward, one reversed) to create a cycle, so that \(G\) must be not be acyclic.
Thus, in all cases that make Statement 3 false, Statement 2 is also false.
Statement 3 implies Statement 4.
Again, we will consider the contrapositive, assuming that Statement 4 is false and proving that Statement 3 is also false.
For Statement 4 to be false, one of the following is true.
- Graph \(G\) is not connected.
- Some edge in \(G\) is not a bridge.
In the first case, \(G\) must contain at least one pair of vertices that is not connected by any path. For the second case, suppose edge \(e\) in \(G\) is a loop. If \(e\) is a loop, then \(G\) contains loops. If \(e\) is not a loop, then it is an edge between a pair of distinct vertices, say \(v\) and \(v'\text{.}\) But then removing \(e\) from \(G\) would leave a subgraph \(G'\) that still contains both \(v\) and \(v'\text{,}\) and which is still connected. So this subgraph (and hence \(G\)) must contain a path between \(v\) and \(v'\) that does not involve \(e\text{.}\) On the other hand, \(v, e, v'\) is also a path between \(v\) and \(v'\text{.}\) So \(v,v'\) is a pair of distinct vertices in \(G\) for which there is more than one path between them.
Thus, in all cases that make Statement 4 false, Statement 3 is also false.
Statement 4 implies Statement 1.
Again, we consider the contrapositive of this logical implication, assuming that Statement 1 is false and proving that Statement 4 is also false. However, since both statements contain the substatement that \(G\) is connected, we will not negate that part.
So assume that \(G\) is connected but contains a proper cycle. We aim to prove that at least one edge in \(G\) is not a bridge. In Activity 16.7.4 , you are asked to prove that none of the edges in the proper cycle that \(G\) contains is a bridge, which will complete the proof.
- Proof of the equivalence of Statement 1, Statement 5, and Statement 6.
-
Statement 1 implies Statement 5.
Assume that \(G\) is a tree. Then it is connected. To prove that the number of edges is \(\vert E \vert = n - 1\text{,}\) we proceed by (strong) induction on \(n\text{,}\) the number of vertices in \(G\text{.}\)For the base base case of \(n = 1\text{,}\) \(\vert E \vert = 0\) is the only possibility, as loops are not allowed in a tree.
Now the induction step. Assume that every tree with \(k \lt n\) vertices has \(k - 1\) edges. Choose some edge in \(G\text{.}\) By Statement 4, removing that edge creates two connected components, \(G_1\) and \(G_2\text{.}\) As \(G\) is acyclic, these connected components are both trees (Statement 2 of Proposition 16.2.1 ). Let \(k_1, k_2\) represent the number of vertices in \(G_1,G_2\text{,}\) respectively, so that \(k_1 + k_2 = n\text{.}\) Since each of \(k_1\) and \(k_2\) must be strictly less than \(n\text{,}\) we may apply our indution hypothesis to each of \(G_1\) and \(G_2\text{,}\) so that \(G_1\) has exactly \(k_1 - 1\) edges and \(G_2\) has exactly \(k_2 - 1\) edges.
Adding up the number of edges in \(G_1\) and \(G_2\text{,}\) along with the single edge in \(G\) that was removed to create these two connected components, we obtain \(\vert E \vert = (k_1 - 1) + (k_2 - 1) + 1 = (k_1 + k_2) - 1 = n - 1, \) as desired.
Statement 5 implies Statement 6.
Consider the contrapositive of this logical implication, assuming that Statement 6 is false and proving that Statement 5 is also false. However, since both statements contain the substatement that \(\vert E \vert = n - 1\text{,}\) we will not negate that part.So assume that \(G\) has \(n - 1\) edges, but contains a proper cycle. We must prove that \(G\) cannot be connected in this case. Choose an edge \(e\) in the proper cycle and create a subgraph \(G'\) by removing \(e\text{.}\) Subgraph \(G'\) now has \(n\) vertices but \(n - 2\) edges, and so by the contrapositive of Theorem 15.3.1 \(G'\) cannot be connected. That means that \(G'\) contains a pair of vertices \(v_1,v_2\) between which no walk exists. If there is a walk between \(v_1,v_2\) in \(G\) but not in \(G'\text{,}\) then that walk must involve the chosen \(e\text{.}\) But then there would be another walk between \(v_1,v_2\) in \(G\) avoiding \(e\) via the rest of the proper cycle containing \(e\text{.}\) And this other walk would be in \(G'\) since it does not involve \(e\text{.}\) Except that we assumed there was no walk between \(v,v'\) in \(G'\text{,}\) hence there also can be no walk between them in \(G\text{.}\) Thus, \(G\) is not connected.
Statement 6 implies Statement 1.
Again, consider the contrapositive of this logical implication, assuming that Statement 1 is false and proving that Statement 6 is also false. However, since both statements contain the substatement that \(G\) is acyclic (part of the definition of tree ), we will not negate that part.So assume that \(G\) is acyclic but not a tree, i.e. that \(G\) is not connected. We must prove that the number of edges in \(G\) is different from \(n - 1\text{,}\) where \(n\) is the number of vertices in \(G\text{.}\) Let \(G_1,G_2,\ldots,G_\ell\) be the connected components of \(G\text{.}\) Now, since \(G\) is assumed acyclic, each of these connected components is a tree (Statement 2 of Proposition 16.2.1 ). We have already proved above that Statement 1 implies Statement 5, so if we write \(k_i\) for the number of vertices in component \(G_i\text{,}\) then we may conclude that the number of edges in component \(G_i\) is \(k_i - 1\text{.}\) As the components make up the entire graph \(G\text{,}\) we may add up the vertices and edges in each component to get the totals in the full graph:
\begin{equation*} n = k_1 + k_2 + \cdots + k_\ell, \end{equation*}
\begin{align*} \vert E \vert & = (k_1 - 1) + (k_2 - 1) + \cdots + (k_\ell - 1) \\ & = (k_1 + k_2 + \cdots + k_\ell) - \ell \\ & = n - \ell \text{.} \end{align*}
Since we assume \(G\) is not connected, we have \(\ell \ge 2\text{,}\) and so \(\vert E \vert \neq n - 1\) as desired.


8.2: Sparse Matrix Formats
8.2.1 List of Lists (LIL)
The List of Lists sparse matrix format (which is, for some unknown reason, abbreviated as LIL rather than LOL) is one of the simplest sparse matrix formats. It is shown schematically in the figure below. Each non-zero matrix row is represented by an element in a kind of list structure called a "linked list". Each element in the linked list records the row number, and the column data for the matrix entries in that row. The column data consists of a list, where each list element corresponds to a non-zero matrix element, and stores information about (i) the column number, and (ii) the value of the matrix element.
Evidently, this format is pretty memory-efficient. The list of rows only needs to be as long as the number of non-zero matrix rows; the rest are omitted. Likewise, each list of column data only needs to be as long as the number of non-zero elements on that row. The total amount of memory required is proportional to the number of non-zero elements, regardless of the size of the matrix itself.
Compared to the other sparse matrix formats which we'll discuss, accessing an individual matrix element in LIL format is relatively slow. This is because looking up a given matrix index \((i,j)\) requires stepping through the row list to find an element with row index \(i\); and if one is found, stepping through the column row to find index \(j\). Thus, for example, looking up an element in a diagonal \(N\times N\) matrix in the LIL format takes \(O(N)\) time! As we'll see, the CSR and CSC formats are much more efficient at element access. For the same reason, matrix arithmetic in the LIL format is very inefficient.
One advantage of the LIL format, however, is that it is relatively easy to alter the "sparsity structure" of the matrix. To add a new non-zero element, one simply has to step through the row list, and either (i) insert a new element into the linked list if the row was not previously on the list (this insertion takes \(O(1)\) time), or (ii) modify the column list (which is usually very short if the matrix is very sparse).
For this reason, the LIL format is preferred if you need to construct a sparse matrix where the non-zero elements are not distributed in any useful pattern. One way is to create an empty matrix, then fill in the elements one by one, as shown in the following example. The LIL matrix is created by the
lil_matrix
function, which is provided by the
scipy.sparse
module.
Here is an example program which constructs a LIL matrix, and prints it:
from scipy import * import scipy.sparse as sp A = sp.lil_matrix((4,5)) # Create empty 4x5 LIL matrix A[0,1] = 1.0 A[1,1] = 2.0 A[1,2] = -1.0 A[3,0] = 6.6 A[3,4] = 1.4 ## Verify the matrix contents by printing it print(A)
When we run the above program, it displays the non-zero elements of the sparse matrix:
(0, 1) 1.0 (1, 1) 2.0 (1, 2) -1.0 (3, 0) 6.6 (3, 4) 1.4
You can also convert the sparse matrix into a conventional 2D array, using the
toarray
method. Suppose we replace the line
print(A)
in the above program with
B = A.toarray() print(B)
The result is:
[[ 0. 1. 0. 0. 0. ] [ 0. 2. -1. 0. 0. ] [ 0. 0. 0. 0. 0. ] [ 6.6 0. 0. 0. 1.4]]
Note: be careful when calling
toarray
in actual code. If the matrix is huge, the 2D array will eat up unreasonable amounts of memory. It is not uncommon to work with sparse matrices with sizes on the order of
, which can take up less an 1MB of memory in a sparse format, but around 80 GB of memory in array format!
8.2.2 Diagonal Storage (DIA)
The Diagonal Storage (DIA) format stores the contents of a sparse matrix along its diagonals. It makes use of a 2D array, which we denote by
data
, and a 1D integer array, which we denote by
offsets
. A typical example is shown in the following figure:
Each row of the
data
array stores one of the diagonals of the matrix, and
offsets[i]
records which diagonal that row of the
data
corresponds to, with "offset 0" corresponding to the main diagonal. For instance, in the above example, row \(0\) of
data
contains the entries
[6.6,0,0,0,0]
), and
offsets[0]
contains the value \(-3\), indicating that the entry \(6.6\) occurs along the \(-3\) subdiagonal, in column \(0\). (The extra elements in that row of
data
lie outside the bounds of the matrix, and are ignored.) Diagonals containing only zero are omitted.
For sparse matrices with very few non-zero diagonals, such as diagonal or tridiagonal matrices, the DIA format allows for very quick arithmetic operations. Its main limitation is that looking up each matrix element requires performing a blind search through the
offsets
array. That's fine if there are very few non-zero diagonals, as
offsets
will be small. But if the number of non-zero diagonals becomes large, performance becomes very poor. In the worst-case scenario of an
anti-diagonal matrix
, element lookup takes \(O(N)\) time!
You can create a sparse matrix in the DIA format, using the
dia_matrix
function, which is provided by the
scipy.sparse
module. Here is an example program:
from scipy import * import scipy.sparse as sp N = 6 # Matrix size diag0 = -2 * ones(N) diag1 = ones(N) A = sp.dia_matrix(([diag1, diag0, diag1], [-1,0,1]), shape=(N,N)) ## Verify the matrix contents by printing it print(A.toarray())
Here, the first input to
dia_matrix
is a tuple of the form
(data, offsets)
, where
data
and
offsets
are arrays of the sort described above. This returns a sparse matrix in the DIA format, with the specified contents (the elements in
data
which lie outside the bounds of the matrix are ignored). In this example, the matrix is tridiagonal with -2 along the main diagonal and 1 along the +1 and -1 diagonals. Running the above program prints the following:
[[-2. 1. 0. 0. 0. 0.] [ 1. -2. 1. 0. 0. 0.] [ 0. 1. -2. 1. 0. 0.] [ 0. 0. 1. -2. 1. 0.] [ 0. 0. 0. 1. -2. 1.] [ 0. 0. 0. 0. 1. -2.]]
Another way to create a DIA matrix is to first create a matrix in another format (e.g. a conventional 2D array), and provide that as the input to
dia_matrix
. This returns a sparse matrix with the same contents, in DIA format.
8.2.3 Compressed Sparse Row (CSR)
The
Compressed Sparse Row
(CSR) format represents a sparse matrix using three arrays, which we denote by
data
,
indices
, and
indptr
. An example is shown in the following figure:
The array denoted
data
stores the values of the non-zero elements of the matrix, in sequential order from left to right along each row, then from the top row to the bottom. The array denoted
indices
records the
column
index for each of these elements. In the above example,
data[3]
stores a value of \(6.6\), and
indices[3]
has a value of \(0\), indicating that a matrix element with value \(6.6\) occurs in column \(0\). These two arrays have the same length, equal to the number of non-zero elements in the sparse matrix.
The array denoted
indptr
(which stands for "index pointer") provides an association between the row indices and the matrix elements, but in an indirect manner. Its length is equal to the number of matrix rows (including zero rows). For each row \(i\), if the row is non-zero,
indptr[i]
records the index
in the
data
and
indices
arrays
corresponding to the first non-zero element on row \(i\). (For a zero row,
indptr
records the index of the next non-zero element occurring in the matrix.)
For example, consider looking up index \((1,2)\) in the above figure. The row index is 1, so we examine
indptr[1]
(whose value is \(1\)) and
indptr[2]
(whose value is \(3\)). This means that the non-zero elements for matrix row \(1\) correspond to indices \(1 \le n < 3\) of the
data
and
indices
arrays. We search
indices[1]
and
indices[2]
, looking for a column index of \(2\). This is found in
indices[2]
, so we look in
data[2]
for the value of the matrix element, which is \(-1.0\).
It is clear that looking up an individual matrix element is very efficient. Unlike the LIL format, where we need to step through a linked list, in the CSR format the
indptr
array lets us to jump straight to the data for the relevant row. For the same reason, the CSR format is efficient for row slicing operations (e.g.,
A[4,:]
), and for matrix-vector products like \(\mathbf{A}\vec{x}\) (which involves taking the product of each matrix row with the vector \(\vec{x}\)).
The CSR format does have several downsides. Column slicing (e.g.
A[:,4]
) is inefficient, since it requires searching through all elements of the
indices
array for the relevant column index. Changes to the sparsity structure (e.g., inserting new elements) are also very inefficient, since all three arrays need to be re-arranged.
To create a sparse matrix in the CSR format, we use the
csr_matrix
function, which is provided by the
scipy.sparse
module. Here is an example program:
from scipy import * import scipy.sparse as sp data = [1.0, 2.0, -1.0, 6.6, 1.4] rows = [0, 1, 1, 3, 3] cols = [1, 1, 2, 0, 4] A = sp.csr_matrix((data, [rows, cols]), shape=(4,5)) print(A)
Here, the first input to
csr_matrix
is a tuple of the form
(data, idx)
, where
data
is a 1D array specifying the non-zero matrix elements,
idx[0,:]
specifies the row indices, and
idx[1,:]
specifies the column indices. Running the program produces the expected results:
(0, 1) 1.0 (1, 1) 2.0 (1, 2) -1.0 (3, 0) 6.6 (3, 4) 1.4
The
csr_matrix
function figures out and generates the three CSR arrays automatically; you don't need to work them out yourself. But if you like, you can inspect the contents of the CSR arrays directly:
>>> A.data array([ 1. , 2. , -1. , 6.6, 1.4]) >>> A.indices array([1, 1, 2, 0, 4], dtype=int32) >>> A.indptr array([0, 1, 3, 3, 5], dtype=int32)
(There is an extra trailing element of \(5\) in the
indptr
array. For simplicity, we didn't mention this in the above discussion, but you should be able to figure out why it's there.)
Another way to create a CSR matrix is to first create a matrix in another format (e.g. a conventional 2D array, or a sparse matrix in LIL format), and provide that as the input to
csr_matrix
. This returns the specified matrix in CSR format. For example:
>>> A = sp.lil_matrix((6,6)) >>> A[0,1] = 4.0 >>> A[2,0] = 5.0 >>> B = sp.csr_matrix(A)
8.2.4 Compressed Sparse Column (CSC)
The
Compressed Sparse Column
(CSC) format is very similar to the CSR format, except that the role of rows and columns is swapped. The
data
array stores non-zero matrix elements in sequential order from top to bottom along each column, then from the left-most column to the right-most. The
indices
array stores
row
indices, and each element of the
indptr
array corresponds to one
column
of the matrix. An example is shown below:
The CSC format is efficient at matrix lookup,
column
slicing operations (e.g.,
A[:,4]
), and vector-matrix products like \(\vec{x}^{\,T} \mathbf{A}\) (which involves taking the product of the vector \(\vec{x}\) with each matrix column). However, it is inefficient for row slicing (e.g.
A[4,:]
), and for changes to the sparsity structure.
To create a sparse matrix in the CSC format, we use the
csc_matrix
function. This is analogous to the
csr_matrix
function for CSR matrices. For example,
>>> from scipy import * >>> import scipy.sparse as sp >>> data = [1.0, 2.0, -1.0, 6.6, 1.4] >>> rows = [0, 1, 1, 3, 3] >>> cols = [1, 1, 2, 0, 4] >>> >>> A = sp.csc_matrix((data, [rows, cols]), shape=(4,5)) >>> A <4x5 sparse matrix of type '<class 'numpy.float64'>' with 5 stored elements in Compressed Sparse Column format>


7.2: Relevance of Race and Ethnicity in the United States
The importance of race and ethnicity is variable both across space and across time. Historically, divisions in the United States along ethnic or racial lines have been the norm. but now these divisions based on race or ethnicity are not as prevalent as they have previously been. From the earliest days of the country and codified in the US Constitution, slavery created a profoundly divided society, particularly between the free, white population and the enslaved Black population. Free people of color provided a small degree of linkage between the groups.
These were not the only divisions in the US, however. The dominant group were people of English descent. In geographical terms, we refer to them as a charter group. The charter group does not refer to the first people to come to a place; they are people with the first effective settlement . This is an academic way of saying that they are the first group with political dominance. English settlers produced laws that furthered their own interests. They promoted their own language (English), religion (Protestant Christianity) and governance. Groups coming in later found themselves in a place where many of the cultural questions had already been answered. The pressure to assimilate in the United States applies to everyone. There can be political pressure; for example, during World War One German Americans largely stopped speaking German. The pressure can be social; for example, young children at school can feel isolated when they cannot speak the majority language. Particularly, the pressure can be economic. Without conforming to general social (majority) norms, it can be difficult to navigate the employment market. A lack of English, unawareness of the norms of formal dress or behavior, or just the inability to recognize social cues can make life difficult for those who have not acculturated.
The charter group also changed. For example, the definitions of “whiteness” and “blackness” have not been historically constant. Consider the history of the United States. Initially, the U.S. population was made largely of Protestant British white people and African black people. Adding people from other places required that definitions be amended.
Would Catholic Italians be considered “White?” In the past, many Americans would have said no. For that matter, neither would the Irish (because of their Catholicism) or Jews (because they aren’t Christian), but over time, these groups were generally included into the white category. Whiteness broadened to include more people. It became less of an ethnic category and more of a racial category.
Definitions of blackness evolved as well. In the American South, there eventually arose a legal framework that defined blackness as having any African ancestry. It would be possible (and relatively common) to be phenotypically white and legally black. Historically, mixed-race Creole people in Louisiana did not consider themselves to be black or white; they were another category altogether.
People attempting to emigrate from Asia, particularly China, were subject to their own set of exclusionary laws, which severely limited their migration to the United States. As late as World War II, it was considered acceptable for the government to intern (imprison) American citizens of Japanese descent over questions of their racial origins and loyalty.
One of the current interesting ethnic questions in the United States is the status of Hispanic people in the existing racial categories. Since Hispanic is not itself a racial category, people within this ethnicity can choose what label they feel is most appropriate. It appears now that Hispanics are identifying themselves as white in the U.S. census. This has an impact on projections for the future U.S. population. If Hispanics identify as white, then the U.S. will remain majority white for quite some time. If they do not, the U.S. will have no racial majority in a few decades.
Although race and ethnicity in the U.S. were largely associated with state- mandated identification, restrictive laws, and onerous obligations, today both race and ethnicity are self-identified for the census. Whereas at one time being Irish could be enough to deny someone employment, now it is a slogan to place on your welcome mat and celebrate once a year in March.
7.2.1 Racial Identifiers
The language used to identify racial groups has changed as well. For example, in broad terms of ethnicity, people of Asian descent who were born in the United States are now referred to as Asian Americans, although the census racial category is still Asian. The term Asian implies a relationship with Asia and no relationship with America. Asian American explicitly ties this group to America.
People who trace their ancestry to Africa have a different problem. This problem is a function of American history. The first census label for this group was simply Black. Over time other labels were used, such as Negro (which means black), and eventually the term African American was adopted. This term is meant to provide a relationship between a population of people and a place of origin. In other words, it explicitly ties a group to their ancestral origin.
Although Native American is used in common speech in the U.S., the Census category is still American Indian, which is not the same as Indian-American (peoples associated with South Asia). The continued use of American Indian is somewhat outside the trend toward more descriptive categories.
7.2.2 Racism in the United States
Although racism and ethnic discrimination are similar, they are not the same thing. Although ethnic markers (generally) diminish over time, physical differences do not.
Exclusionary racial policies existed in the United States from the very beginning and have continued beyond the Jim Crow era of the twentieth century. From the US Constitution that counted slaves as 3/5 of a person to restrictive housing covenants in the 1960s, the country has had a history of racism that did not end in the Civil Rights era. This exclusion has not solely been limited to African Americans. Many groups have been subject to racist laws and acts. The indigenous people of the United States were not fully considered citizens until 1924. In the past, voting rights, access to housing and even union membership had racialized politics directed at many marginalized groups.
This is not to say that ethnically-based discrimination does not exist. Such discrimination has been prevalent in United States history, but it tends to subside as the host population absorbs the immigrant population.
7.2.3 Housing
Historically, ethnic groups tended to live near one another in spatially contiguous areas. Many cities have a Chinatown or Little Italy. These are known as ethnic enclaves . There are many reasons why groups cluster; some reasons are voluntary and some are not. In the United States, it was not uncommon for cities to restrict where African American citizens could live. These restrictions were either through the force of law, or through unwritten behavioral norms that resisted renting or selling houses to African American families outside of certain areas. This residential, spatial segregation was accompanied with educational, social, and economic segregation. African American communities were often known as ghettos , places where a certain population is forced to live. The word ghetto is older than the United States itself. Ghetto was an Italian name for the area that Jews were forced to live in. Although the word is Italian, the idea of forcing minority populations to live in designated areas has unfortunately had wide historical appeal. Legal housing segregation ended in the United States in 1968, but behaviors change more slowly than laws.
Many ethnic communities have arisen from less coercive means. There are numerous reasons that an ethnic community would choose to live close together. Mutual support networks, the ability to develop schools and businesses catering to their own needs, a sense of safety, and the ability to retain their own cultural connections are examples of positive reasons. Institutionalized poverty, marginalized political representation, and active discrimination are negative reasons.
7.2.4 Environmental Justice
One of the spatial manifestations of racism and ethnic discrimination is the difference in levels of political representation. Another one is the location of unpleasant environmental activities. Landfills and airports tend to be built in places inhabited by less-powerful groups, while dominant groups rarely if ever have to organize to prevent such things being built in their neighborhoods. Some groups find their economic situations limited by underfunded schools or inadequate infrastructure. The idea that different groups should have access to decent places to live called environmental justice .
7.2.5 Ethnic diversity in the United States
Like all predominantly immigrant countries, the United States is ethnically diverse, but the range of ethnicities has varied over time as new groups arrive and previous groups acculturate and eventually assimilate. A male of Italian descent in the United States will sometimes just say, “I’m Italian.” This may be a person who speaks no Italian, isn’t Catholic, and never been in Italy in his entire life. What then, does this statement mean? It just signals an historic connection with an ethnicity, even if the connection has faded over time. This isn’t to single out Italian- Americans. Generally, as groups assimilate, their distinctive ethnic markers fade. Comparing Polish-Americans with Mexican-Americans may involve people who speak the same language (English), have the same Catholic religion, and live verysimilar lifestyles. The label has faded to a marker, with food being the one of the last cultural elements.
7.2.6 Foodways
One of the ways groups demonstrate ethnicity is through food. One of the most obvious hallmarks of the arrival of an ethnicity into the United States, or any other country, is the diffusion of a food from the group of origin. Pizza in the United States, curry in the United Kingdom, and doner kebab in Germany all exemplify the degree to which a food brought by immigrants can reach the status of adopted national cuisine. Food is also the cultural element that is most accessible to outsiders. Foodways are used to construct a spatial sense of one location as a reflection of the entire world.
Foodways refer to the types of food that people eat, the ways they are prepared, and the cultural factors that surround and contextualize the food. Food is the most resilient cultural artifact. In countries undergoing language unification, foods can define ethnic groups. In mostly monolingual countries like the United States, foods may indicate geographical origins or social class. Food is easily bought, tried and accepted, or rejected. As such, it is the most accessible cultural element.
In many ways, the consumption of a food and its production have been divorced from its roots by the modern restaurant industry and international food conglomerates. Americans have eaten foods they consider Chinese or Mexican for generations, while few know the histories of said foods. Questions of whether or not a food is authentic are difficult to answer when the cooks in a restaurant are of a completely different ethnicity from the stated cuisine.
We can compare foodways between places and groups. Quantities of food, the ratio of prepared foods, and consumption of tobacco and alcohol all help us get inside the lives of people in different places, at different states of technological development, and different socioeconomic classes.
7.2.7 The Ethnic Landscape
Urban ethnic landscapes are often immediately recognizable. Signs in other languages advertising exotic products, ethnic architecture, and even local tourism reveal the ethnic fabric of a place. Most people in the US do not live in large cities with obvious ethnic architecture. The majority of Americans, including many ethnicities, live in the suburbs and smaller towns. Instead of obvious population clusters, ethnic populations here can be widely dispersed. Instead of living within walking distance of their local store or religious structure, people will simply drive to such a place. Ethnicity has sprawled along with the rest of America. Waves of migration to U.S. cities and suburbs have created landscapes of tremendous ethnic difference embedded in architectural homogeneity.
7.2.8 Ethnic Festivals and the Idealized Homeland
One of the ways that ethnicities represent themselves is through festivals. People wear traditional clothing, play music from the old country, eat food previously reserved for holidays, dance the old dances, and promote their culture to others. Festivals are a way of reproducing a sense of home in emigrant communities. They are also a way of keeping children participating in activities that would otherwise forget.
Places represented in ethnic festivals in the United States are often not representative of those places now. Traditional Czech clothing at a Kolache Festival in Oklahoma represents a place/time that no longer exists, except perhaps to market “Czech-ness” to tourists.