section-3-4.xml

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

<section xml:id="Rational-Exponents"   xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Rational Exponents</title>

<subsection><title>Powers of the Form <m>a^{m/n}</m></title>
<p>
    In the last section, we considered powers of the form <m>a^{1/n}</m>, such as <m>x^{1/3}</m> and <m>x^{-1/4}</m>, and saw that <m>a^{1/n}</m> is equivalent to the root <m>\sqrt[n]{a}</m>. What about other fractional exponents? What meaning can we attach to a power of the form <m>a^{m /n}</m>?
</p>
<p>
    Consider the power <m>x^{3/2}</m>. Notice that the exponent <m>\dfrac{3}{2}= 3(\dfrac{1}{2})</m>, and thus by the third law of exponents, we can write
    <me>(x^{1/2})^3 = x^{(1/2)^3} = x^{3/2}</me>
    In other words, we can compute <m>x^{3/2}</m> by first taking the square root of <m>x</m> and then cubing the result. For example,
    \begin{align*}
        100^{3/2} \amp = (\alert{100^{1/2}})^3 \amp\amp \text{Take the square root of 100.}\\
        \amp= \alert{10^3} = 1000  \amp\amp \text{Cube the result.}
    \end{align*}
    We will define fractional powers only when the base is a positive number.
</p>

<assemblage><title>Rational Exponents</title><p>
<me>
    a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}\text{, }a \gt 0\text{, } n \ne 0
</me></p>
</assemblage>

<p>
    To compute <m>a^{m/n}</m>, we can compute the <m>n</m>th root first, or the <m>m</m>th power, whichever is easier. For example,
    <me>8^{2/3} = \left(8^{1/3}\right)^2 = 2^2 = 4</me>
    or
    <me>8^{2/3} = \left(8^2\right)^{1/3} = 64^{1/3} = 4</me>
</p>

<example xml:id="example-rational-exponent">
    <ol label="a">
        <li><p>\begin{align*}
            81^{3/4} \amp = \left(81^{1/4}\right)^3 \\
            \amp = 3^3 = 27
            \end{align*}
        </p></li>
        <li><p>\begin{align*} 
            -27^{5/3} \amp = -\left(27^{1/3}\right)^5 \\
            \amp = -3^5 = -243
            \end{align*}
        </p></li>
        <li><p>\begin{align*}
            27^{-2/3} \amp = \frac{1}{\left(27^{1/3}\right)^2} \\
            \amp = \frac{1}{3^2}= \frac{1}{9}
            \end{align*}
        </p></li>
        <li><p>
            \begin{align*}
            5^{3/2} \amp = \left(5^{1/2}\right)^3 \\
            \amp \approx (2.236)^3 \approx 11.180
            \end{align*}
        </p></li>
    </ol>
</example>

<note><p>
    You can verify all the calculations in <xref ref="example-rational-exponent" text="type-global"/> on your calculator. For example, to evaluate <m>81^{3/4}</m>, key in
</p>
<p style="text-align:center">
    81 <c>^</c> ( 3 <c>÷</c> 4 ) <c>ENTER</c>
</p>
<p>or simply</p>
<p halign="center">
    81 <c>^</c> 0.75 <c>ENTER</c>
</p></note>

<exercise xml:id="exercise-evaluate-rational-exponent"><statement>
    Evaluate each power.
    <ol label="a">
        <li><p><m> 32^{-3/5}</m></p></li>
        <li><p><m> -81^{1.25}</m></p></li>
    </ol>
</statement>
<answer><p><ol cols="2" label="a">
    <li><p><m>\dfrac{1}{8} </m></p></li>
    <li><p><m>-243</m></p></li>
</ol></p></answer>
</exercise>

</subsection>

<subsection><title>Power Functions</title><idx>power functions</idx>
<p>
    The graphs of power functions <m>y = x^{m/n}</m>, where <m>m/n</m> is positive, are all increasing for <m>x\ge 0</m>. If <m>m/n \gt 1</m>, the graph is concave up. If <m>0 \lt m/n \lt 1</m>, the graph is concave down. Some examples are shown in <xref ref="fig-power-functions" text="type-global"/>.
</p>
<figure xml:id="fig-power-functions"><caption></caption><image source="images/fig-power-functions"  width="40%"><description>graphs of power functions</description></image></figure>
<p>
    Perhaps the single most useful piece of information a scientist can have about an animal is its metabolic rate. The metabolic rate is the amount of energy the animal uses per unit of time for its usual activities, including locomotion, growth, and reproduction. The basal metabolic rate, or BMR, sometimes called the resting metabolic rate, is the minimum amount of energy the animal can expend in order to survive.
</p>

<example xml:id="example-Kleiber-rule">
<p>
    A revised form of Kleiber's rule states that the basal metabolic rate for many groups of animals is given by 
    <me>B(m) = 70m^{0.75}</me>
    where <m>m</m> is the mass of the animal in kilograms and the BMR is measured in kilocalories per day.
</p>
<ol label="a">
    <li><p>Calculate the BMR for various animals whose masses are 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>Bat</cell>
            <cell>Squirrel</cell>
            <cell>Raccoon</cell>
            <cell>Lynx</cell>
            <cell>Human</cell>
            <cell>Moose</cell>
            <cell>Rhinoceros</cell>
        </row>
        <row>
            <cell>Weight (kg)</cell>
            <cell><m>0.1</m></cell>
            <cell><m>0.6</m></cell>
            <cell><m>8</m></cell>
            <cell><m>30</m></cell>
            <cell><m>70</m></cell>
            <cell><m>360</m></cell>
            <cell><m>3500</m></cell>
        </row>
        <row>
            <cell>BMR (kcal/day)</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 a graph of Kleiber’s rule for <m>0 \lt m \le 400</m>.</p></li>
    <li><p>Do larger species eat more or less, relative to their body mass, than smaller ones?</p></li>
</ol>
<solution>
<ol label="a">
    <li><p>We evaluate the function for the values of <m>m</m> given. For example, to calculate the BMR of a bat, we compute
    <me>B(0.1) = 70(0.1)^{0.75} = 12.1</me>
    A bat expends, and hence must consume, at least <m>12</m> kilocalories per day. We evaluate the function to complete the rest of the table.</p><p>
    <sidebyside><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell>Animal</cell>
            <cell>Bat</cell>
            <cell>Squirrel</cell>
            <cell>Raccoon</cell>
            <cell>Lynx</cell>
            <cell>Human</cell>
            <cell>Moose</cell>
            <cell>Rhinoceros</cell>
        </row>
        <row>
            <cell>Weight (kg)</cell>
            <cell><m>0.1</m></cell>
            <cell><m>0.6</m></cell>
            <cell><m>8</m></cell>
            <cell><m>30</m></cell>
            <cell><m>70</m></cell>
            <cell><m>360</m></cell>
            <cell><m>3500</m></cell>
        </row>
        <row>
            <cell>BMR (kcal/day)</cell>
            <cell><m>12</m></cell>
            <cell><m>48</m></cell>
            <cell><m>333</m></cell>
            <cell><m>897</m></cell>
            <cell><m>1694</m></cell>
            <cell><m>5785</m></cell>
            <cell><m>31,853</m></cell>
        </row>
    </tabular></sidebyside>
    </p></li>
    <li><p>We plot the data from the table to obtain the graph in <xref ref="fig-bat-to-rhino" text="type-global"/>.
    <figure xml:id="fig-bat-to-rhino"><caption></caption><image source="images/fig-bat-to-rhino"  width="45%"><description>BMR vs mass for bat to rhino</description></image></figure>
    </p></li>
    <li><p>If energy consumption were proportional to body weight, the graph would be a straight line. But because the exponent in Kleiber’s rule, <m>\dfrac{3}{4}</m>, is less than <m>1</m>, the graph is concave down, or bends downward. Therefore, larger species eat less than smaller ones, relative to their body weight. </p></li>
</ol>
</solution>
</example>

<exercise xml:id="exercise-power-function"><statement>
<ol label="a">
    <li><p>Complete the table of values for the function <m>f (x) = x^{-3/4}</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>0.2</m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
            <cell><m>5</m></cell>
            <cell><m>8</m></cell>
            <cell><m>10</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>
        </row>
    </tabular></sidebyside>
</p></li>
    <li><p>Sketch the graph of the function.</p></li>
</ol>
</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>0.2</m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
            <cell><m>5</m></cell>
            <cell><m>8</m></cell>
            <cell><m>10</m></cell>
        </row>
        <row>
            <cell><m>f (x)</m></cell>
            <cell><m>5.623</m></cell>
            <cell><m>3.344</m></cell>
            <cell><m>1.682</m></cell>
            <cell><m>1</m></cell>
            <cell><m>0.595</m></cell>
            <cell><m>0.299</m></cell>
            <cell><m>0.210</m></cell>
            <cell><m>0.178</m></cell>
        </row>
    </tabular></p></li>
    <li><p><image source="images/fig-in-ex-ans-3-4-2"  width="45%"><description>power function</description></image> </p></li>
</ol></p></answer>
</exercise>

</subsection>

<subsection><title>More about Scaling</title>
<p>
    In <xref ref="example-Kleiber-rule" text="type-global"/> we saw that large animals eat less than smaller ones, relative to their body weight. This is because the scaling exponent in Kleiber's rule is less than <m>1</m>. For example, let <m>s</m> represent the mass of a squirrel. The mass of a moose is then <m>600s</m>, and its metabolic rate is
    \begin{align*}
    B(600s) \amp = 70(600s)^{0.75} \\
    \amp = 600^{0.75} \cdot 70s^{0.75} = 121B(s)
    \end{align*}
    or <m>121</m> times the metabolic rate of the squirrel. Metabolic rate scales as <m>k^{0.75}</m>, compared to the mass of the animal.
</p>
<p>
    In a famous experiment in the 1960s, an elephant was given LSD. The dose was determined from a previous experiment in which a <m>2.6</m>-kg cat was given <m>0.26</m> gram of LSD. Because the elephant weighed <m>2970</m> kg, the experimenters used a direct proportion to calculate the dose for the elephant:
    <me>\frac{0.26 \text{ g}}{2.6 \text{ kg}}= \frac{x \text{ g}}{2970 \text{ kg}}</me>
    and arrived at the figure <m>297</m> g of LSD. Unfortunately, the elephant did not survive the experiment.
</p>

<example xml:id="example-Kleiber-elephant-LSD-dose">
<p>
    Use Kleiber's rule and the dosage for a cat to estimate the corresponding dose for an elephant.
</p>
<solution><p>
    If the experimenters had taken into account the scaling exponent of <m>0.75</m> in metabolic rate, they would have used a smaller dose. Because the elephant weighs <m>\dfrac{2970}{2.6}</m> , or about <m>1142</m> times as much as the cat, the dose would be <m>1142^{0.75} = 196</m> times the dosage for a cat, or about <m>51</m> grams.
</p></solution>
</example>

<exercise xml:id="exercise-LSD-human-vs-elephant-dose"><statement>
    A human being weighs about <m>70</m> kg, and <m>0.2</m> mg of LSD is enough to induce severe psychotic symptoms. Use these data and Kleiber's rule to predict what dosage would produce a similar effect in an elephant.
</statement>
<answer><p>About <m>3.3</m> mg</p></answer>
</exercise>

</subsection>

<subsection><title>Radical Notation</title><idx>radical notation</idx>
    <p>
        Because <m>a^{1/n} = \sqrt[n]{a}</m>, we can write any power with a fractional exponent in radical form as follows.
    </p>

<assemblage><title>Rational Exponents and Radicals</title><p>
    <me>
        a^{m/n} = \sqrt[n]{a^m} =\left( \sqrt[n]{a}\right)^m
    </me></p>
</assemblage>

<example xml:id="example-rational-exponent-to-radicals">
    <ol label="a">
        <li><p><m>125^{4/3} = \sqrt[3]{125^4} \text{ or } \left(\sqrt[3]{125}\right)^4</m></p></li>
        <li><p><m>x^{0.4} = x^{2/5} = \sqrt[5]{x^2}</m></p></li>
        <li><p><m>6w^{-3/4} = \dfrac{6}{\sqrt[4]{w^3}}</m></p></li>
    </ol>
</example>

<exercise xml:id="exercise-rational-exponent-to-radical"><statement>
Write each expression in radical notation.
<ol label="a">
    <li><p><m>5t^{1.25}</m></p></li>
    <li><p><m> 3m^{-5/3}</m></p></li>
</ol>
</statement>
<answer><p><ol cols="2" label="a">
    <li><p><m>5\sqrt[4]{t^5} </m></p></li>
    <li><p><m>\dfrac{3}{\sqrt[3]{m^5} } </m></p></li>
</ol></p></answer>
</exercise>

<p>
    Usually, we will want to convert from radical notation to fractional exponents, since exponential notation is easier to use.
</p>

<example xml:id="example-radical-to-exponential">
<ol label="a">
    <li><p><m>\sqrt{x^5} = x^{5/2}</m></p></li>
    <li><p><m> 5 \sqrt[4]{p^3} = 5p^{3/4}</m></p></li>
    <li><p><m> \dfrac{3}{\sqrt[5]{t^2}}= 3t^{-2/5}</m></p></li>
    <li><p><m> \sqrt[3]{2y^2} = \left(2y^2\right)^{1/3} = 2^{1/3} y^{2/3}</m></p></li>
</ol>
</example>

<exercise xml:id="exercise-radical-to-exponential"><statement>
<p>    Convert to exponential notation.</p>
    <ol label="a">
        <li><p><m>\sqrt[3]{6w^2}</m></p></li>
        <li><p><m>  \sqrt[4]{\dfrac{v^3}{s^5}}</m></p></li>
    </ol>
</statement>
<answer><p><ol cols="2" label="a">
    <li><p><m>6^{1/3} w^{2/3} </m></p></li>
    <li><p><m>v^{3/4} s^{-5/4} </m></p></li>
</ol></p></answer>
</exercise>

</subsection>

<subsection><title>Operations with Rational Exponents</title>
<p>
    Powers with rational exponents—positive, negative, or zero—obey the laws of exponents, which we discussed in <xref ref="Variation" text="type-global"/>. You may want to review those laws before studying the following examples.
</p>

<example xml:id="example-operations-with-rational-exponents">
<ol label="a">
    <li><p>\begin{align*}
        \frac{7^{0.75}}{7^{0.5}}\amp= 7^{0.75-0.5} = 7^{0.25}
         \amp\amp \text{Apply the second law of exponents.}
         \end{align*}
    </p></li>
    <li><p>\begin{align*}
        v \cdot v^{-2/3}\amp= v^{1+(-2/3)}
        \amp\amp \text{Apply the first law of exponents.}\\
        \amp  = v^{1/3}
        \end{align*}
    </p></li>
    <li><p>\begin{align*}
        \left(x^8\right)^{0.5}\amp= x^{8(0.5))} = x^4
        \amp\amp \text{Apply the third law of exponents.}
        \end{align*}
    </p></li>
    <li><p>\begin{align*}
        \frac{\left(5^{1/2}y^2\right)^2}{\left(5^{2/3} y\right)^3}  \amp= \frac{5y^4}{5^2 y^3}
        \amp\amp \text{Apply the fourth law of exponents.}\\
        \amp = \frac{y^{4-3}}{5^{2-1}}=\frac{y}{5}
        \amp\amp \text{Apply the second law of exponents.}\\
        \end{align*}
    </p></li>
</ol>
</example>

<exercise xml:id="exercise-operations-on-rational-exponents"><statement>
    Simplify by applying the laws of exponents.
    <ol label="a">
        <li><m> x^{1/3}\left(x + x^{2/3}\right)</m></li>
        <li><m> \dfrac{n^{9/4}}{4n^{3/4}}</m></li>
    </ol>
</statement>
<answer><p><ol label="a">
    <li><p><m>x^{4/3}+x </m></p></li>
    <li><p><m>\dfrac{n^{3/2}}{4} </m></p></li>
</ol></p></answer>
</exercise>

</subsection>

<subsection><title>Solving Equations</title>
<p>
    According to the third law of exponents, when we raise a power to another power, we multiply the exponents together. In particular, if the two exponents are reciprocals, then their product is <m>1</m>. For example,
    <me>\left(x^{2/3}\right)^{3/2} = x^{(2/3) (3/2)} = x^1 = x</me>
    This observation can help us to solve equations involving fractional exponents. For instance, to solve the equation
    <me>x^{2/3} = 4</me>
    we raise both sides of the equation to the reciprocal power, <m>3/2</m>. This gives us
    \begin{align*}
    \left(x^{2/3}\right)^{3/2} \amp = 4^{3/2} \\
    x \amp = 8
    \end{align*}
    The solution is <m>8</m>.
</p>

<example xml:id="example-equation-with-rational-exponents">
<p>Solve <m>(2x + 1)^{3/4} = 27</m>.</p>
<solution>
<p>
    We raise both sides of the equation to the reciprocal power, <m>\frac{4}{3}</m>.
    \begin{align*}
    \left[(2x + 1)^{3/4}\right]^{4/3} \amp= 27^{4/3} 
    \amp\amp  \text{Apply the third law of exponents.} \\
    2x + 1 \amp = 81 \amp\amp \text{Solve as usual.} \\
    x \amp = 40
    \end{align*}
</p>
</solution>
</example>

<exercise xml:id="exercise-equation-with-rational-exponent"><statement>
    <p>
        Solve the equation <m>3.2z^{0.6} - 9.7 = 8.7</m>. Round your answer to two decimal places.
    </p>
</statement>
<hint>
    <p>Isolate the power.</p>
    <p>Raise both sides to the reciprocal power.</p>
</hint>
<answer><p><m>18.45</m></p></answer>
</exercise>


<investigation><title>Vampire Bats</title>
<statement><p>
    Small animals such as bats cannot survive for long without eating. The graph in <xref ref="fig-vampire-bats" text="type-global"/> shows how the weight, <m>W</m>, of a typical vampire bat decreases over time until its next meal, until the bat reaches the point of starvation. The curve is the graph of the function
    <me>W(h) = 130.25h^{-0.126}</me>
    where <m>h</m> is the number of hours since the bat’s most recent meal. (Source: Wilkinson, 1984)
</p>
<figure xml:id="fig-vampire-bats"><caption></caption><image source="images/fig-vampire-bats"  width="50%"><description>bat weight vs survival time</description></image></figure>

<ol>
    <li>Use the graph to estimate answers to the following questions: How long can the bat survive after eating until its next meal? What is the bat’s weight at the point of starvation?</li>
    <li>Use the formula for <m>W(h)</m> to verify your answers.</li>
    <li>Write and solve an equation to answer the question: When the bat's weight has dropped to <m>90</m> grams, how long can it survive before eating again?</li>
    <li><p>Complete the table showing the number of hours since the bat last ate when its weight has dropped to the given values.</p><p>
    <sidebyside><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell>Weight (grams)</cell>
            <cell><m>97.5</m></cell>
            <cell><m>92.5</m></cell>
            <cell><m>85</m></cell>
            <cell><m>80</m></cell>
        </row>
        <row>
            <cell>Hours since eating</cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
            <cell><m></m></cell>
        </row>
        <row>
            <cell>Point on graph</cell>
            <cell><m>A</m></cell>
            <cell><m>B</m></cell>
            <cell><m>C</m></cell>
            <cell><m>D</m></cell>
        </row>
    </tabular></sidebyside></p>
    </li>
    <li>
        Compute the slope of the line segments from point <m>A</m> to point <m>B</m>, and from point <m>C</m> to point <m>D</m>. (See <xref ref="fig-vampire-bats2" text="type-global"/>.) Include units in your answers.
        <figure xml:id="fig-vampire-bats2"><caption></caption><image source="images/fig-vampire-bats2"  width="50%"><description>bat weight vs survival time</description></image></figure>
    </li>
    <li>
        What happens to the slope of the curve as <m>h</m> increases? What does this tell you about the concavity of the curve?
    </li>
    <li>
        Suppose a bat that weighs <m>80</m> grams consumes <m>5</m> grams of blood. How many hours of life does it gain? Suppose a bat that weighs <m>97.5</m> grams gives up a meal of <m>5</m> grams of blood. How many hours of life does it forfeit?
    </li>
    <li>
        Vampire bats sometimes donate blood (through regurgitation) to other bats that are close to starvation. Suppose a bat at point <m>A</m> on the curve donates <m>5</m> grams of blood to a bat at point <m>D</m>. Explain why this strategy is effective for the survival of the bat community.
    </li>

</ol></statement>
</investigation>

</subsection>
<xi:include href="./summary-3-4.xml" /> <!-- summary  -->
<xi:include href="./section-3-4-exercises.xml" /> <!-- exercises  -->

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