section-3-3.xml

<?xml version="1.0" encoding="UTF-8" ?>
<!-- <mathbook><book> -->

<section xml:id="Roots-and-Radicals"   xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Roots and Radicals</title><introduction>
<p>
    In <xref ref="Integer-Exponents" text="type-global"/> we saw that inverse variation can be expressed as a power function by using negative exponents. We can also use exponents to denote square roots and other radicals.
</p></introduction>
<subsection><title><m>n</m>th Roots</title><idx>nth roots</idx>
<p>
    Recall that <m>s</m> is a square root of <m>b</m> if <m>s^2 = b</m>, and <m>s</m> is a cube root of <m>b</m> if <m>s^3 = b</m>. In a similar way, we can define the fourth, fifth, or sixth root of a number. For instance, the fourth root of <m>b</m> is a number <m>s</m> whose fourth power is <m>b</m>. In general, we make the following definition.
</p>
<assemblage><title><m>n</m>th Roots</title>
<p>
    <m>s</m> is called an <term> <m>n</m>th root of <m>b</m></term> if <m>s^n = b</m>.
</p></assemblage>
<p>
   We use the symbol <m>\sqrt[n]{b}</m> to denote the <m>n</m>th root of <m>b</m>. An expression of the form <m>\sqrt[n]{b}</m> is called a <term>radical</term><idx>radical</idx>, <m>b</m> is called the <term>radicand</term><idx>radicand</idx>, and <m>n</m> is called the <term>index of the radical</term><idx>index of the radical</idx>. 
</p>

<example xml:id="example-nth-roots">
    <ol label="a" cols="2">
        <li><m>\sqrt[4]{81} = 3</m> because <m>3^4 = 81</m></li>
        <li><m>\sqrt[5]{32} = 2</m> because <m>2^5 = 32</m></li>
        <li><m>\sqrt[6]{64} = 2</m> because <m>2^6 = 64</m></li>
        <li><m>\sqrt[4]{1} = 1</m> because <m>1^4 = 1</m></li>
        <li><m>\sqrt[5]{100,000} = 10</m> because <m>10^5 = 100,000</m></li>
    </ol>
</example>

<exercise xml:id="exercise-nth-roots"><statement>
    <p>Evaluate each radical.
        <ol label="a" cols="2">
            <li><m>\sqrt[4]{16}</m></li>
            <li><m>\sqrt[5]{243}</m></li>
        </ol>
    </p>
</statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>2</m></p></li>
    <li><p><m>3</m></p></li>
</ol></p></answer>
</exercise>
</subsection>



<subsection><title>Exponential Notation for Radicals</title>
<p>
    A convenient notation for radicals uses fractional exponents. Consider the expression <m>9^{1/2}</m>. What meaning can we attach to an exponent that is a fraction? The third law of exponents says that when we raise a power to a power, we multiply the exponents together:
    <me>\left(x^a\right)^b = x^{ab}</me>
    Therefore, if we square the number <m>9^{1/2}</m>, we get
    <me>\left(9^{1/2}\right)^2 = 9^{(1/2)(2)} = 9^1 = 9</me>
    Thus, <m>9^{1/2}</m> is a number whose square is <m>9</m>. But this means that <m>9^{1/2}</m> is a square root of <m>9</m>, or
    <me>9^{1/2} =\sqrt{9} = 3</me>
    In general, any nonnegative number raised to the <m>1/2</m> power is equal to the positive square root of the number, or
    <me>a^{1/2} =\sqrt{a}</me>
</p>

<example xml:id="example-exponential-notation">
    <ol label="a" cols="2">
        <li><m>25^{1/2} = 5</m></li>
        <li><m>-25^{1/2} = -5</m></li>
        <li><m>(-25)^{1/2}</m> is not a real number.</li>
        <li><m>0^{1/2} = 0</m></li>
    </ol>
</example>

<exercise xml:id="exercise-exponential-notation"><statement>
    <p>Evaluate each power.
    <ol label="a" cols="4">
        <li><m>4^{1/2}</m></li>
        <li><m>4^{-2}</m></li>
        <li><m>4^{-1/2}</m></li>
        <li><m>\left(\dfrac{1}{4}\right)^{1/2}</m></li>
    </ol></p>
</statement>
<answer><p><ol label="a" cols="4">
    <li><p><m>2</m></p></li>
    <li><p><m>\dfrac{1}{16} </m></p></li>
    <li><p><m>\dfrac{1}{2} </m></p></li>
    <li><p><m>\dfrac{1}{2} </m></p></li>
</ol></p></answer>
</exercise>

<p>
    The same reasoning works for roots with any index. For instance, <m>8^{1/3}</m> is the cube root of <m>8</m>, because
    <me>\left(8^{1/3}\right)^3 = 8^{(1/3)(3)} = 8^1 = 8</me>
    In general, we make the following definition for fractional exponents.
</p>

<assemblage><title>Exponential Notation for Radicals</title><p>
    For any integer <m>n \ge 2</m> and for <m>a \ge 0</m>,
    <me>a^{1/n} = \sqrt[n]{a}</me>
</p></assemblage>

<example xml:id="example-exponential-notation2" cols="2">
    <ol label="a">
        <li><m>81^{1/4} = \sqrt[4]{81} = 3</m></li>
        <li><m>125^{1/3} = \sqrt[3]{125} = 5</m></li>
    </ol>
</example>

<warning><statement>
<p>
    Note that 
    <me>25^{1/2} \ne \frac{1}{2}(25) ~~ \text{ and } ~~ 125^{1/3} \ne \frac{1}{3}(125)</me>
    An exponent of <m>\dfrac{1}{2}</m> denotes the square root of its base, and an exponent of <m>\dfrac{1}{3}</m> denotes the cube root of its base.
</p></statement></warning>

<exercise xml:id="exercise-exponential-notation2"><statement>
    <p>
        Write each power with radical notation, and then evaluate.
        <ol label="a" cols="2">
            <li><m> 32^{1/5}</m></li>
            <li><m> 625^{1/4}</m></li>
        </ol>
    </p>
</statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\sqrt[5]{32}= 2</m></p></li>
    <li><p><m>\sqrt[4]{625}= 5</m></p></li>
</ol></p></answer>
</exercise>

<p>
    Of course, we can use decimal fractions for exponents as well. For example,
    <me>\sqrt{a} = a^{1/2} = a^{0.5} ~~\text{ and } ~~ \sqrt[4]{a} = a^{1/4} = a^{0.25}</me>
</p>

<example xml:id="example-decimal-exponential-notation">
    <ol label="a" cols="2">
        <li><m>100^{0.5} = \sqrt{100} = 10</m></li>
        <li><m>16^{0.25} = \sqrt[4]{16} = 2</m></li>
    </ol>
</example>

<exercise xml:id="exercise-decimal-exponential-notation"><statement>
    <p>Write each power with radical notation, and then evaluate.
        <ol label="a" cols="2">
            <li><m>100,000^{0.2}</m></li>
            <li><m>81^{0.25}</m></li>
        </ol>
    </p>
</statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\sqrt[5]{100,000}=10 </m></p></li>
    <li><p><m>\sqrt[4]{81}=3 </m></p></li>
</ol></p></answer>
</exercise>  

</subsection>



<subsection><title>Irrational Numbers</title>
<p>
    What about <m>n</m>th roots such as <m>\sqrt{23}</m> and <m>5^{1/3}</m> that cannot be evaluated easily? These are examples of <term>irrational numbers</term><idx>irrational numbers</idx>. We can use a calculator to obtain decimal approximations for irrational numbers. For example, you can verify that
    <me>\sqrt{23} \approx 4.796 ~ \text{ and } ~ 5^{1/3}\approx 1.710</me>
    It is not possible to write down an exact decimal equivalent for an irrational number, but we can find an approximation to as many decimal places as we like.
</p>

<warning>
<p>
    The following keying sequence for evaluating the irrational number <m>7^{1/5}</m> is incorrect:
</p>
<p halign="center">
    7 <c>^</c> 1  <c>÷</c> 5  <c>ENTER</c>
</p>
<p>
    You can check that this sequence calculates <m>\dfrac{7^1}{5}</m>, instead of <m>7^{1/5}</m>. Recall that according to the order of operations, powers are computed before multiplications or divisions. We must enclose the exponent <m>1/5</m> in parentheses and enter
</p>
<p>
    7 <c>^</c> ( 1 <c>÷</c> 5 ) <c>ENTER</c>   
</p>
<p>
    Or, because <m>\frac{1}{5}= 0.2</m>, we can enter
</p>
<p>
    7 <c>^</c> 0.2 <c>ENTER</c>
</p>
</warning>  

</subsection>



<subsection><title>Working with Fractional Exponents</title>
<p>
    Fractional exponents simplify many calculations involving radicals. You should learn to convert easily between exponential and radical notation. Remember that a negative exponent denotes a reciprocal.
</p>

<example xml:id="example-radical-to-exponential-notation">
    <p>Convert each radical to exponential notation.
        <ol label="a" cols="2">
            <li><m>\sqrt[3]{12} = 12^{1/3}</m></li>
            <li><m>\sqrt[4]{2y} = (2y)^{1/4} \text{ or } (2y)^{0.25}</m></li>
        </ol>
    </p>
</example>

<exercise xml:id="exercise-radical-to-exponential-notation"><statement>
    <p>
        Convert each radical to exponential notation.
        <ol label="a" cols="2">
            <li><m>\dfrac{1}{\sqrt[5]{ab}}</m></li>
            <li><m>3\sqrt[6]{w}</m></li>
        </ol>
    </p>
</statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>(ab)^{-1/5} </m></p></li>
    <li><p><m>3w^{1/6} </m></p></li>
</ol></p></answer>
</exercise>

<example xml:id="example-exponetial-to-radical-notation"><p>
    Convert each power to radical notation.
    <ol label="a" cols="2">
        <li><m>5^{1/2} = \sqrt{5}</m></li>
        <li><m>x^{0.2} = \sqrt[5]{x}</m></li>
        <li><m>2x^{1/3} = 2 \sqrt[3]{x}</m></li>
        <li><m>8a^{-1/4} = \dfrac{8}{\sqrt[4]{a}}</m></li>
    </ol>
</p></example>

<note><p>
    In <xref ref="example-exponetial-to-radical-notation"/>d, note that the exponent <m>-1/4</m> applies only to <m>a</m>, not to <m>8a</m>.
</p></note>

<exercise xml:id="exercise-convert-exponential-radical"><statement><p>
    <ol label="a">
        <li>Convert <m>\dfrac{3}{\sqrt[4]{2x} } </m> to exponential notation.</li>
        <li>Convert <m>-5b^{0.125}</m> to radical notation.</li>
    </ol>
</p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>3(2x)^{-1/4} </m></p></li>
    <li><p><m>-5\sqrt[8]{b} </m></p></li>
</ol></p></answer>
</exercise>

</subsection>



<subsection><title>Using Fractional Exponents to Solve Equations</title>
<p>
    In Chapter 2, we learned that raising to powers and taking roots are inverse operations, that is, each operation undoes the effects of the other. This relationship is especially easy to see when the root is denoted by a fractional exponent. For example, to solve the equation 
    <me>x^4 = 250</me>
    we would take the fourth root of each side. But instead of using radical notation, we can raise both sides of the equation to the power <m>\dfrac{1}{4}</m>:
    \begin{align*}
    \left(x^4\right)^{1/4} \amp= 250^{1/4} \\
    x \amp \approx 3.98
    \end{align*}
    The third law of exponents tells us that <m>\left(x^a\right)^b = x^{ab}</m>, so <m>\left(x^4\right)^{1/4} = x^{(1/4)(4)} = x^1</m>. In general, to solve an equation involving a power function <m>x^n</m>, we first isolate the power, then raise both sides to the exponent <m>\dfrac{1}{n}</m>.
</p>

<example xml:id="example-star-luminosity">
    <p>
        For astronomers, the mass of a star is its most important property, but it is also the most difficult to measure directly. For many stars, their luminosity, or brightness, varies roughly as the fourth power of the mass.
    </p>
    <ol label="a">
        <li>Our Sun has luminosity <m>4 \times 1026</m> watts and mass <m>2 \times 1030</m> kilograms. Because the numbers involved are so large, astronomers often use these solar constants as units of measure: The luminosity of the Sun is <m>1</m> solar luminosity, and its mass is <m>1</m> solar mass. Write a power function for the luminosity, <m>L</m>, of a star in terms of its mass, <m>M</m>, using units of solar mass and solar luminosity.</li>
        <li>The star Sirius is <m>23</m> times brighter than the Sun, so its luminosity is <m>23</m> solar luminosities. Estimate the mass of Sirius in units of solar mass.</li>
    </ol>
<solution>
    <ol label="a">
        <li>Because <m>L</m> varies as the fourth power of <m>M</m>, we have
        <me>L = kM^4</me>
        Substituting the values of <m>L</m> and <m>M</m> for the Sun (namely, <m>L = 1</m> and <m>M = 1</m>), we find
        <me>1 = k(1)^4</me>
        so <m>k = 1</m> and <m>L = M^4</m>.</li>
        <li>We substitute the luminosity of Sirius, <m>L = 23</m>, to get
        <me>23 = M^4</me>
        To solve the equation for <m>M</m>, we raise both sides to the <m>\dfrac{1}{4}</m> power.
        \begin{align*}
        (23)^{1/4} \amp = \left(M^4\right)^{1/4} \\
        2.1899 \amp = M
        \end{align*}
        The mass of Sirius is about <m>2.2</m> solar masses, or about <m>2.2</m> times the mass of the Sun.</li>
    </ol>
</solution>
</example>

<exercise xml:id="exercise-spherical-tank"><statement><p>
    A spherical fish tank in the lobby of the Atlantis Hotel holds about <m>905</m> cubic feet of water. What is the radius of the fish tank?
</p></statement>
<answer><p>About <m>6</m> feet</p></answer>
</exercise>

</subsection>

<subsection><title>Power Functions</title>
<p>
    The basic functions <m>y = \sqrt{x}</m> and <m>y = \sqrt[3]{x}</m> are power functions of the form <m>f (x) = x^{1/n}</m>, and the graphs of all such 
    functions have shapes similar to those two, depending on whether the index of the root is even or odd. </p>

    <p><xref ref="fig-graphs-of-even-and-odd-roots" text="type-global"/>a shows the graphs of <m>y = x^{1/2}</m>, <m>y = x^{1/4}</m>, and <m>y = x^{1/6}</m>. <xref ref="fig-graphs-of-even-and-odd-roots" text="type-global"/>b shows the graphs of <m>y = x^{1/3}</m>, <m>y = x^{1/5}</m>, and <m>y = x^{1/7}</m>.</p>

<figure xml:id="fig-graphs-of-even-and-odd-roots"><caption></caption><image source="images/fig-graphs-of-even-and-odd-roots"  width="85%"><description>graph of even roots and graph of odd roots</description></image></figure>

<p>
    We cannot take an even root of a negative number. (See <xref ref="Roots-of-Negative-Numbers" text="type-global"/> "A Note on Roots of Negative Numbers" at the end of this section.) Hence, if <m>n</m> is even, the domain of <m>f (x) = x^{1/n}</m> is restricted to nonnegative real numbers, but if <m>n</m> is odd, the domain of <m>f (x) = x^{1/n}</m> is the set of all real numbers.
</p>

<p>
    We will also encounter power functions with negative exponents. For example, an animal's heart rate is related to its size or mass, with smaller animals generally having faster heart rates. The heart rates of mammals are given approximately by the power function
    <me>H(m) = km^{-1/4}</me>
    where <m>m</m> is the animal's mass and <m>k</m> is a constant.
</p>

<example xml:id="example-mammal-mass-HR">
    <p>A typical human male weighs about <m>70</m> kilograms and has a resting heart rate of <m>70</m> beats per minute.
    <ol label="a">
        <li>Find the constant of proportionality, <m>k</m>, and write a formula for <m>H(m)</m>.</li>
        <li><p>Fill in the table with the heart rates of the mammals whose masses are given.</p><p>
        <sidebyside><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
                <row bottom="minor">
                    <cell>Animal</cell>
                    <cell>Shrew</cell>
                    <cell>Rabbit</cell>
                    <cell>Cat</cell>
                    <cell>Wolf</cell>
                    <cell>Horse</cell>
                    <cell>Polar bear</cell>
                    <cell>Elephant</cell>
                    <cell>Whale</cell>
                </row>
                <row>
                    <cell>Mass (kg)</cell>
                    <cell><m>0.004</m></cell>
                    <cell><m>2</m></cell>
                    <cell><m>4</m></cell>
                    <cell><m>80</m></cell>
                    <cell><m>300</m></cell>
                    <cell><m>600</m></cell>
                    <cell><m>5400</m></cell>
                    <cell><m>70,000</m></cell>
                </row>
                <row>
                    <cell>Heart rate</cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                    <cell><m></m></cell>
                </row>
        </tabular></sidebyside></p></li>
        <li>Sketch a graph of <m>H</m> for masses up to <m>6000</m> kilograms.</li>
        </ol>
    </p>
<solution>
<ol label="a">
    <li>We substitute <m>H = 70</m> and <m>m = 70</m> into the equation; then solve for <m>k</m>.
    \begin{align*}
    70 \amp = k · 70^{-1/4} \\
    k \amp= \frac{70}{70^{-1/4}} \\
    \amp= 70^{5/4}\approx 202.5
    \end{align*}
    Thus, <m>H(m) = 202.5m^{-1/4}</m>.</li>
    <li><p>We evaluate the function <m>H</m> for each of the masses given in the table.</p><p>
            <sidebyside><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
                <row bottom="minor">
                    <cell>Animal</cell>
                    <cell>Shrew</cell>
                    <cell>Rabbit</cell>
                    <cell>Cat</cell>
                    <cell>Wolf</cell>
                    <cell>Horse</cell>
                    <cell>Polar bear</cell>
                    <cell>Elephant</cell>
                    <cell>Whale</cell>
                </row>
                <row>
                    <cell>Mass (kg)</cell>
                    <cell><m>0.004</m></cell>
                    <cell><m>2</m></cell>
                    <cell><m>4</m></cell>
                    <cell><m>80</m></cell>
                    <cell><m>300</m></cell>
                    <cell><m>600</m></cell>
                    <cell><m>5400</m></cell>
                    <cell><m>70,000</m></cell>
                </row>
                <row>
                    <cell>Heart rate</cell>
                    <cell><m>805</m></cell>
                    <cell><m>170</m></cell>
                    <cell><m>143</m></cell>
                    <cell><m>68</m></cell>
                    <cell><m>49</m></cell>
                    <cell><m>41</m></cell>
                    <cell><m>24</m></cell>
                    <cell><m>12</m></cell>
                </row>
        </tabular></sidebyside></p>
    </li>
    <li>We plot the points in the table to obtain the graph shown in <xref ref="fig-HR-vs-mass" text="type-global"/>.
    <figure xml:id="fig-HR-vs-mass"><caption></caption><image source="images/fig-HR-vs-mass"  width="50%"><description>heart rate vs mass</description></image></figure>
    </li>
</ol>
</solution>
</example>


<p>
    Many properties relating to the growth of plants and animals can be described by power functions of their mass. The study of the relationship between the growth rates of different parts of an organism, or of organisms of similar type, is called <term>allometry</term><idx>allometry</idx>. An equation of the form
    <me>\text{variable} = k(\text{mass})^p</me>
    used to describe such a relationship is called an <term>allometric equation</term><idx>allometric equation</idx>.
</p>
<p>
    Of course, power functions can be expressed using any of the notations we have discussed. For example, the function in <xref ref="example-mammal-mass-HR" text="type-global"/> can be written as
    <me>H(m) = 202.5m^{-1/4} \text{ or } H(m) = 202.5m^{-0.25} \text{ or } H(m) = \frac{202.5}{\sqrt[4]{m}}</me>
</p>

<exercise xml:id="exercise-power-function"><statement><p>
    <ol label="a">
        <li><p>Complete the table of values for the power function <m>f (x) = x^{-1/2}</m>.</p><p>
<sidebyside><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>0.1</m></cell>
            <cell><m>0.25</m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
            <cell><m>4</m></cell>
            <cell><m>8</m></cell>
            <cell><m>10</m></cell>
            <cell><m>20</m></cell>
            <cell><m>200</m></cell>
        </row>
        <row>
            <cell><m>f(x)</m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
    </tabular></sidebyside>
       </p></li>
        <li><p>Sketch the graph of <m>y = f (x)</m>.</p></li>
        <li><p>Write the formula for <m>f (x)</m> with a decimal exponent, and with radical notation.</p></li>
    </ol>
</p></statement>
<answer><p><ol label="a">
    <li><p>
    <tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>0.1</m></cell>
            <cell><m>0.25</m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
            <cell><m>4</m></cell>
            <cell><m>8</m></cell>
            <cell><m>10</m></cell>
            <cell><m>20</m></cell>
            <cell><m>200</m></cell>
        </row>
        <row>
            <cell><m>f(x)</m></cell>
            <cell><m>3.2</m></cell>
            <cell><m>2</m></cell>
            <cell><m>1.4</m></cell>
            <cell><m>1</m></cell>
            <cell><m>0.71</m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>0.35</m></cell>
            <cell><m>0.32</m></cell>
            <cell><m>0.22</m></cell>
            <cell><m>0.1</m></cell>
        </row>
    </tabular>
    </p></li>
    <li><p><image source="images/fig-in-ex-ans-3-3-8"  width="30%"><description>power function</description></image> </p></li>
    <li><p><m>f (x) = x^{-0.5}</m>, <m>f(x)=\dfrac{1}{\sqrt{x}} </m></p></li>
</ol></p></answer>
</exercise>

</subsection>




<subsection><title>Solving Radical Equations</title>
<p>
    A <term>radical equation</term><idx>radical equation</idx> is one in which the variable appears under a square root or other radical. The radical may be denoted by a fractional exponent. For example, the equation
    <me>5x^{1/3} = 32</me>
    is a radical equation because <m>x^{1/3} = \sqrt[3]{x}</m>. To solve the equation, we first isolate the power to get
    <me>x^{1/3} = 6.4</me>
    Then we raise both sides of the equation to the reciprocal of <m>\dfrac{1}{3}</m>, or <m>3</m>.
    \begin{align*}
    \left(x^{1/3}\right)^3 \amp = 6.4^3 \\
    x \amp = 262.144
    \end{align*}
</p>

<example xml:id="example-car-brakes">
<p>
    When a car brakes suddenly, its speed can be estimated from the length of the skid marks it leaves on the pavement. A formula for the car’s speed, in miles per hour, is <m>v = f (d) = (24d)^{1/2}</m>, where the length of the skid marks, <m>d</m>, is given in feet.
</p>
<ol label="a">
    <li>If a car leaves skid marks <m>80</m> feet long, how fast was the car traveling when the driver applied the brakes?</li>
    <li>How far will a car skid if its driver applies the brakes while traveling <m>80</m> miles per hour?</li>
</ol>
<solution>
<ol label="a">
    <li>To find the velocity of the car, we evaluate the function for <m>d = \alert{80}</m>.
    \begin{align*}
     v\amp= (24 \cdot \alert{80})^{1/2} \\
     \amp = (1920)^{1/2} \\
     \amp \approx 43.8178046
    \end{align*}
    The car was traveling at approximately <m>44</m> miles per hour.</li>
    <li>We would like to find the value of <m>d</m> when the value of <m>v</m> is known. We substitute <m>v = \alert{80}</m> into the formula and solve the equation
    <me>\alert{80} = (24d)^{1/2} ~~ \text{ Solve for }d.</me>
    Because <m>d</m> appears to the power <m>\frac{1}{2}</m>, we first square both sides of the equation to get
    \begin{align*}
    80^2 \amp = \left((24d)^{1/2}\right)^2 \amp\amp\text{Square both sides.}\\
    6400 \amp = 24d \amp\amp\text{Divide by }24.\\
    266.\overline{6} \amp = d
    \end{align*}
    You can check that this value for <m>d</m> works in the original equation. Thus, the car will skid approximately <m>267</m> feet. A graph of the function <m>v = (24d)^{1/2}</m> is shown in <xref ref="fig-velocity-vs-braking-distance" text="type-global"/>, along with the points corresponding to the values in parts (a) and (b).
    <figure xml:id="fig-velocity-vs-braking-distance"><caption></caption><image source="images/fig-velocity-vs-braking-distance"  width="60%"><description>velocity vs braking distance</description></image></figure>
 </li> </ol>
</solution></example>

<p>
    Thus, we can solve an equation where one side is an <m>n</m>th root of <m>x</m> by raising both sides of the equation to the <m>n</m>th power. We must be careful when raising both sides of an equation to an even power, since extraneous solutions may be introduced. However, because most applications of power functions deal with positive domains only, they do not usually involve extraneous solutions.
</p>

<exercise xml:id="exercise-mammal-mass-HR"><statement><p>
    In <xref ref="example-mammal-mass-HR" text="type-global"/>, we found the heart-rate function, <m>H(m) = 202.5m^{-1/4}</m>. What would be the mass of an animal whose heart rate is <m>120</m> beats per minute?
</p></statement>
<answer><p><m>81</m> kg</p></answer>
</exercise>

</subsection>


<subsection xml:id="Roots-of-Negative-Numbers"><title>A Note on Roots of Negative Numbers</title>
<p>
    You already know that <m>\sqrt{-9}</m> is not a real number, because there is no real number whose square is <m>-9</m>. Similarly, <m>\sqrt[4]{-16}</m> is not a real number, because there is no real number <m>r</m> for which <m>r^4 = -16</m>. (Both of these radicals are <term>complex numbers</term><idx>complex numbers</idx>. Complex numbers are discussed in Chapter 7.) In general, we cannot find an even root (square root, fourth root, and so on) of a negative number.
</p>
<p>
    On the other hand, every positive number has two even roots that are real numbers. For example, both <m>3</m> and <m>-3</m> are square roots of <m>9</m>. The symbol <m>\sqrt{9}</m> refers only to the positive, or <term>principal root</term>, of <m>9</m>. If we want to refer to the negative square root of <m>9</m>, we must write <m>-\sqrt{9} = -3</m>. Similarly, both <m>2</m> and <m>-2</m> are fourth roots of <m>16</m>, because <m>2^4 = 16</m> and <m>(-2)^4 = 16</m>. However, the symbol <m>\sqrt[4]{16}</m> refers to the principal, or positive, fourth root only. Thus,
    <me>\sqrt[4]{16} = 2 ~~ \text{ and } ~~ -\sqrt[4]{16} = -2</me>
</p>
<p>
    Things are simpler for odd roots (cube roots, fifth roots, and so on). Every real number, whether positive, negative, or zero, has exactly one real-valued odd root. For example,
    <me>\sqrt[5]{32} = 2 ~~ \text{ and } ~~ \sqrt[5]{-32} = -2</me>
</p>
<p>
    Here is a summary of our discussion.
</p>

<assemblage><title>Roots of Real Numbers</title><p>
<ol>
    <li>Every positive number has two real-valued roots, one positive and one negative, if the index is even.</li>
    <li>A negative number has no real-valued root if the index is even.</li>
    <li>Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd.</li>
</ol></p>
</assemblage>

<example xml:id="example-roots-of-negatives">
    <ol label="a">
        <li><m>\sqrt[4]{-625}</m> is not a real number.</li>
        <li><m> - \sqrt[4]{625} = -5</m></li>
        <li><m> \sqrt[5]{-1} = -1</m></li>
        <li><m> \sqrt[4]{-1}</m> is not a real number.</li>
    </ol>
</example>

<p>
    The same principles apply to powers with fractional exponents. Thus
    <me>(-32)^{1/5} = -2</me>
    but <m>(-64)^{1/6}</m> is not a real number. On the other hand,
    <me>-64^{1/6} = -2</me>
    because the exponent <m>1/6</m> applies only to <m>64</m>, and the negative sign is applied after the root is computed.
</p>

<exercise xml:id="exercise-roots-of-negatives"><statement><p>
    Evaluate each power, if possible.
    <ol label="a" cols="2">
        <li><m>-81^{1/4}</m></li>
        <li><m> (-81)^{1/4}</m></li>
        <li><m> -64^{1/3}</m></li>
        <li><m> (-64)^{1/3}</m></li>
    </ol>
</p></statement>
<answer><p><ol label="a" cols="4">
    <li><p><m>-3</m></p></li>
    <li><p>undefined</p></li>
    <li><p><m>-4</m></p></li>
    <li><p><m>-4</m></p></li>
</ol></p></answer>
</exercise>


</subsection>
<xi:include href="./summary-3-3.xml" /> <!-- summary  -->
<xi:include href="./section-3-3-exercises.xml" /> <!-- exercises  -->

</section> 
<!-- </book>  </mathbook> -->