In the last section, we considered powers of the form \(a^{1/n}\text{,}\) such as \(x^{1/3}\) and \(x^{-1/4}\text{,}\) and saw that \(a^{1/n}\) is equivalent to the root \(\sqrt[n]{a}\text{.}\) What about other fractional exponents? What meaning can we attach to a power of the form \(a^{m /n}\text{?}\)

Consider the power \(x^{3/2}\text{.}\) Notice that the exponent \(\dfrac{3}{2}= 3(\dfrac{1}{2})\text{,}\) and thus by the third law of exponents, we can write

\(\dfrac{4}{3} = 4\left(\dfrac{1}{3}\right)\text{,}\) and in general \(\dfrac{p}{q} = p\left(\dfrac{1}{q}\right)\text{.}\)

True

False

To compute \((25^{1/2})^3\text{,}\) we first take the square root of 25, then cube the result.

True

False

The notation \(16^{3/4}\) means to multipy 16 by \(\dfrac{3}{4}\text{.}\)

True

False

Answer1.

\(\text{True}\)

Answer2.

\(\text{True}\)

Answer3.

\(\text{True}\)

Answer4.

\(\text{False}\)

Solution.

True

True

True

False

A fractional exponent represents a power and a root. The denominator of the exponent is the root, and the numerator of the exponent is the power. We will define fractional exponents only when the base is a positive number.

You can verify all the calculations in Example 6.3.2 on your calculator. For example, to evaluate \(81^{3/4}\text{,}\) key in

81 ^ ( 3 ÷ 4 ) ENTER

or simply

81 ^ 0.75 ENTER

Caution6.3.4.

When computing \(81^{3/4}\) on a calculator, do not forget the parentheses around the exponent, \(\dfrac{3}{4}\text{.}\) The keying sequence

\(\qquad\qquad 81\)^\(3\)÷ENTER gives us the value of \(\dfrac{81^3}{4}\text{,}\) or \(132,860.25\text{.}\) The parentheses tell the calculator to include all of the quotient \(3 \div 4\) in the exponent, not just the 3.

Checkpoint6.3.5.Practice 1.

Evaluate each power.

\(32^{-3/5}=\)

\(-81^{1.25}=\)

Answer1.

\({\frac{1}{8}}\)

Answer2.

\(-243\)

Solution.

\(\displaystyle \dfrac{1}{8}\)

\(\displaystyle -243\)

Subsection6.3.2Power Functions

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.

Example6.3.6.

A revised form of Kleiber’s rule states that the basal metabolic rate for many groups of animals is given by

A bat expends, and hence must consume, at least 12 kilocalories per day. We evaluate the function to complete the rest of the table. The values of BMR are rounded to the nearest whole number.

Animal

Bat

Squirrel

Raccoon

Lynx

Human

Moose

Rhinoceros

Weight (kg)

\(0.1\)

\(0.6\)

\(8\)

\(30\)

\(70\)

\(360\)

\(3500\)

BMR (kcal/day)

\(12\)

\(48\)

\(333\)

\(897\)

\(1694\)

\(5785\)

\(31,853\)

We plot the data from the table to obtain the graph below.

If energy consumption were proportional to body weight, the graph would be a straight line. But this graph is concave down, or bends downward. Larger species eat less than smaller ones, relative to their body weight. For example, a moose weighs 600 times as much as a squirrel, but its energy requirement is only 121 times the squirrel’s.

Checkpoint6.3.7.QuickCheck 2.

True or false.

The graph of \(f(x)=70x^{3/4}\) is concave down.

True

False

Energy consumption is not directly proportional to body weight.

True

False

An animal twice as heavy as a squirrel uses twice as much energy as a squirrel.

True

False

The graph of \(f(x)=70x^{3/4}\) passes through the origin.

True

False

Answer1.

\(\text{True}\)

Answer2.

\(\text{True}\)

Answer3.

\(\text{False}\)

Answer4.

\(\text{True}\)

Solution.

True

True

False

True

Checkpoint6.3.8.Practice 2.

Complete the table of values for the function \(f (x) = x^{-3/4}\text{.}\)

\(x\)

\(0.1\)

\(0.2\)

\(0.5\)

\(1\)

\(f (x)\)

\(x\)

\(2\)

\(5\)

\(8\)

\(10\)

\(f (x)\)

Sketch the graph of the function.

Answer1.

\(5.62341\)

Answer2.

\(3.3437\)

Answer3.

\(1.68179\)

Answer4.

\(1\)

Answer5.

\(0.594604\)

Answer6.

\(0.29907\)

Answer7.

\(0.210224\)

Answer8.

\(0.177828\)

Solution.

\(x\)

\(0.1\)

\(0.2\)

\(0.5\)

\(1\)

\(f (x)\)

\(5.623\)

\(3.344\)

\(1.682\)

\(1\)

\(x\)

\(2\)

\(5\)

\(8\)

\(10\)

\(f (x)\)

\(0.595\)

\(0.299\)

\(0.210\)

\(0.178\)

A graph is below.

Subsection6.3.3Radical Notation

Because \(a^{1/n} = \sqrt[n]{a}\text{,}\) we can write any power with a fractional exponent in radical form as follows.

\(x^{3/5}\) represents the cube root of \(x\) to the fifth power.

True

False

We can simplify \(2x^{4/3}\) as \(\sqrt[3]{2x^4}\text{.}\)

True

False

The notation \(a^{-1/4}\) means \(-a^4\text{.}\)

True

False

We can simplify \(5b^{-3/2}\) as \(\dfrac{1}{5b^{3/2}}\text{.}\)

True

False

Answer1.

\(\text{False}\)

Answer2.

\(\text{False}\)

Answer3.

\(\text{False}\)

Answer4.

\(\text{False}\)

Solution.

False

False

False

False

Subsection6.3.4Operations with Rational Exponents

Powers with rational exponents—positive, negative, or zero—obey the laws of exponents, which we discussed in Section 6.1. You may want to review those laws before studying the following examples.

Example6.3.14.

\begin{align*}
\frac{7^{0.75}}{7^{0.5}}\amp= 7^{0.75-0.5} = 7^{0.25}
\amp\amp \blert{\text{Apply the second law of exponents.}}
\end{align*}

\begin{align*}
v \cdot v^{-2/3}\amp= v^{1+(-2/3)}
\amp\amp \blert{\text{Apply the first law of exponents.}}\\
\amp = v^{1/3}
\end{align*}

\begin{align*}
\left(x^8\right)^{0.5}\amp= x^{8(0.5))} = x^4
\amp\amp \blert{\text{Apply the third law of exponents.}}
\end{align*}

\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 \blert{\text{Apply the fourth law of exponents.}}\\
\amp = \frac{y^{4-3}}{5^{2-1}}=\frac{y}{5}
\amp\amp \blert{\text{Apply the second law of exponents.}}
\end{align*}

Checkpoint6.3.15.Practice 5.

Simplify by applying the laws of exponents.

\(x^{1/3}\left(x + x^{2/3}\right)=\)

\(\dfrac{n^{9/4}}{4n^{3/4}}=\)

Answer1.

\(x^{\frac{4}{3}}+x\)

Answer2.

\(\frac{n^{\frac{3}{2}}}{4}\)

Solution.

\(\displaystyle x^{4/3}+x\)

\(\displaystyle \dfrac{n^{3/2}}{4}\)

Subsection6.3.5Solving Equations

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 \(1\text{.}\) For example,

\begin{equation*}
\left(x^{2/3}\right)^{3/2} = x^{(2/3) (3/2)} = x^1 = x
\end{equation*}

This observation can help us to solve equations involving fractional exponents. For instance, to solve the equation

\begin{equation*}
x^{2/3} = 4
\end{equation*}

we raise both sides of the equation to the reciprocal power, \(3/2\text{.}\) This gives us

We raise both sides of the equation to the reciprocal power, \(\dfrac{4}{3}\text{.}\)

\begin{align*}
\left[(2x + 1)^{3/4}\right]^{4/3} \amp= 27^{4/3}
\amp\amp \blert{\text{Apply the third law of exponents.}}\\
2x + 1 \amp = 81 \amp\amp \blert{\text{Solve as usual.}}\\
x \amp = 40
\end{align*}

Checkpoint6.3.17.Practice 6.

Solve the equation \(3.2z^{0.6} - 9.7 = 8.7\text{.}\) Round your answer to two decimal places.

Answer:

Hint.

\(\blert{\text{Isolate the power.}}\)

\(\blert{\text{Raise both sides to the reciprocal power}}.\)

Answer.

\(\left(\frac{18.4}{3.2}\right)^{\frac{1}{0.6}}\)

Solution.

\(18.45\)

Checkpoint6.3.18.QuickCheck 3.

True or false.

The first step in solving an equation with a fractional exponent is to isolate the power.

True

False

Then we raise both sides to the reciprocal of the exponent.

True

False

We can use the laws of exponents to simplify expressions involving fractional exponents.

True

False

The reciprocal of 0.6 is \(\dfrac{1}{0.6}\text{.}\)

True

False

Answer1.

\(\text{True}\)

Answer2.

\(\text{True}\)

Answer3.

\(\text{True}\)

Answer4.

\(\text{True}\)

Solution.

True

True

True

True

Investigation6.3.1.Inflating a Balloon.

If you blow air into a balloon, what do you think will happen to the air pressure inside the balloon as it expands? Here is what two physics books have to say:

“The greater the pressure inside, the greater the balloon’s volume.” Jay Boleman, Physics, a Window on Our World

“Contrary to the process of blowing up a toy balloon, the pressure required to force air into a bubble decreases with bubble size.” Francis Sears, Mechanics, Heat, and Sound

On the basis of these two quotations and your own intuition, sketch a graph of pressure as a function of the diameter of the balloon. Describe your graph: Is it increasing or decreasing? Is it concave up (bending upward) or concave down (bending downward)?

Two high school students, April Leonardo and Tolu Noah, decided to see for themselves how the pressure inside a balloon changes as the balloon expands. Using a column of water to measure pressure, they collected the following data while blowing up a balloon. Graph their data on the grid below.

Diameter (cm)

Pressure (cm \(\text{H}_2\)O)

\(5.7\)

\(60.6\)

\(7.3\)

\(57.2\)

\(8.2\)

\(47.9\)

\(10.7\)

\(38.1\)

\(12\)

\(37.1\)

\(14.6\)

\(31.9\)

\(17.5\)

\(28.1\)

\(20.5\)

\(26.4\)

\(23.5\)

\(28\)

\(25.2\)

\(31.4\)

\(26.1\)

\(34\)

\(27.5\)

\(37.2\)

\(28.4\)

\(37.9\)

\(29\)

\(40.7\)

\(30\)

\(43.3\)

\(30.6\)

\(46.6\)

\(31.3\)

\(50\)

\(32.2\)

\(61.9\)

Describe the graph of April and Tolu’s data. Does the graph confirm the predictions of the physics books?

As the diameter of the balloon increases from 5 cm to 20 cm, the pressure inside decreases. Can we find a function that describes this portion of the graph? Here is some information:

Pressure is the force per unit area exerted by the balloon on the air inside, or \(P=\dfrac{F}{A}\text{.}\)

The balloon is spherical, so its surface area, \(A\text{,}\) is given by \(A=\pi d^2\).

Because the force increases as the balloon expands, we will try a power function of the form \(F=kd^p\text{,}\) where \(k\) and \(p\) are constants, to see if it fits the data. Combine the three equations, \(P=\dfrac{F}{A},~A=\pi d^2~\text{,}\) and \(~F=kd^p\text{,}\) to express \(P\) as a power function of \(d\text{.}\)

Graph the function \(P=211 d^{-0.7}\) on the same grid with the data. Do the data support the hypothesis that \(P\) is a power function of \(d\text{?}\)

What is the value of the exponent \(p\) in \(~F=kd^p\text{?}\)

Exercises6.3.6Problem Set 6.3

Warm Up

1.

Combining powers and roots

Start with 8, square it, and then take the cube root of the result. That is:

Do you get the same answers for parts (a) and (b)? You should! Now try the same thing with some irrational numbers. Use your calculator, and round your answers to two decimal places.

For Problems 33 and 34, solve the equations, and locate the corresponding points on the graphs in Problems 31 and 32.

33.

\(\displaystyle x^{4/3}=4 \)

\(\displaystyle x^{3/2}=6\)

Answer.

\(\displaystyle 2.83\)

\(\displaystyle 3.30\)

34.

\(\displaystyle x^{3/4}=4 \)

\(\displaystyle x^{2/3}=6\)

Applications

35.

During a flu epidemic in a small town, health officials estimate that the number of people infected \(t\) days after the first case was discovered is given by

Graph the function \(I(t)\) on your calculator, and verify your answer to part (b) on your graph.

Answer.

\(t\)

\(5\)

\(10\)

\(15\)

\(20\)

\(I(t)\)

\(131\)

\(199\)

\(254\)

\(302\)

20 days

36.

The research division of an advertising firm estimates that the number of people who have seen their ads \(t\) days after the campaign begins is given by the function

How long will it be before 75,000 people have seen the ads?

Graph the function \(n(t)\) on your calculator, and verify your answer to part (b) on your graph.

37.

In the 1970s, Jared Diamond studied the number of bird species on small islands near New Guinea. He found that larger islands support a larger number of different species, according to the formula

\begin{equation*}
S = 15.1A^{0.22}
\end{equation*}

where \(S\) is the number of species on an island of area \(A\) square kilometers. (Source: Chapman and Reiss, 1992)

Fill in the table.

\(A\)

\(10\)

\(100\)

\(1000\)

\(5000\)

\(10,000\)

\(S\)

\(\hphantom{000000} \)

\(\hphantom{000000} \)

\(\hphantom{000000} \)

\(\hphantom{000000} \)

\(\hphantom{000000} \)

Graph the function on your calculator for \(0\lt A\le 10,000\text{.}\)

How many species of birds would you expect to find on Manus Island, with an area of 2100 square kilometers? On Lavongai, whose area is 1140 square kilometers?

How large must an island be in order to support 200 different species of bird?

Answer.

\(A\)

\(10\)

\(100\)

\(1000\)

\(5000\)

\(10,000\)

\(S\)

\(25 \)

\(42 \)

\(69 \)

\(98 \)

\(115 \)

81, 71

126,000 sq km

38.

The climate of a region has a great influence on the types of animals that can survive there. Extreme temperatures create difficult living conditions, so the diversity of wildlife decreases as the annual temperature range increases. Along the west coast of North America, the number of species of mammals, \(M\text{,}\) is approximately related to the temperature range, \(R\text{,}\) (in degrees Celsius) by the function

\begin{equation*}
M = f(R) = 433.8R^{-0.742}
\end{equation*}

(Source: Chapman and Reiss, 1992)

Graph the function for temperature ranges up to \(30\degree\)C.

How many species would you expect to find in a region where the temperature range is \(10\degree \)C? Label the corresponding point on your graph.

If 50 different species are found in a certain region, what temperature range would you expect the region to experience? Label the corresponding point on your graph.

Evaluate the function to find \(f(9)\text{,}\)\(f(10)\text{,}\)\(f(19)\text{,}\) and \(f(20)\text{.}\) What do these values represent? Calculate the change in the number of species as the temperature range increases from \(9\degree\)C to \(10\degree\)C and from \(19\degree\)C to \(20\degree\)C. Which \(1\degree\) increase results in a greater decrease in diversity? Explain your answer in terms of slopes on your graph.

39.

The table at right shows the exponent, \(p\text{,}\) in the allometric equation

\begin{gather*}
\text{variable} = k (\text{body mass})^p
\end{gather*}

for some variables related to mammals. (Source: Chapman and Reiss, 1992)

Variable

Exponent, \(p\)

Home range size

\(1.26\)

Lung volume

\(1.02\)

Brain mass

\(0.70\)

Heart rate

\(-0.25\)

The average body mass of a dolphin is about 140 kilograms, twice the body mass of an average human male. Using the allometric equation above, calculate the ratio of the brain mass of a dolphin to that of a human.

A good-sized brown bear weighs about 280 kilograms, twice the weight of a dolphin. Calculate the ratio of the brain mass of a brown bear to that of a dolphin.

Use a ratio to compare the heartbeat frequencies of a dolphin and a human, and those of a brown bear and a dolphin.

Answer.

1.62

1.62

0.84; 0.84

40.

A bicycle ergometer is used to measure the amount of power generated by a cyclist. The scatterplot shows how long an athlete was able to sustain variouslevels of power output. The curve is the graph of

\begin{equation*}
y = 500x^{0.29}
\end{equation*}

which approximately models the data. (Source: Alexander, 1992)

As represented by the graph, which variable is the input, and which is the output?

The athlete maintained 650 watts of power for 40 seconds. What power output does the equation predict for 40 seconds?

The athlete maintained 300 watts of power for 10 minutes. How long does the equation predict that power output can be maintained?

In 1979 a remarkable pedal-powered aircraft called the Gossamer Albatross was successfully flown across the English Channel. The flight took 3 hours. According to the equation, what level of power can be maintained for 3 hours?

The Gossamer Albatross needed 250 watts of power to keep it airborne. For how long can 250 watts be maintained, according to the equation?

41.

Birds’ eggs typically lose 10%–20% of their mass during incubation. The embryo metabolizes lipid during growth, and this process releases water vapor through the porous shell. The incubation time for birds’ eggs is a function of the mass of the egg and has been experimentally determined as

Combine the functions \(I(m)\) and \(O(m)\) to calculate the total amount of oxygen taken in by the egg during its incubation.

Use your result from part (a) to explain why total oxygen consumption per unit mass is approximately inversely proportional to incubation time. (Oxygen consumption is a reliable indicator of metabolic rate, and it is reasonable that incubation time should be inversely proportional to metabolic rate.)

Predict the oxygen consumption per gram of a herring gull’s eggs, given that their incubation time is 26 days. (The actual value is 11 milliliters per day.)

43.

Kepler’s law gives a relation between the period, \(p\text{,}\) of a planet’s revolution and its average distance, \(a\text{,}\) from the Sun:

\begin{equation*}
p^2 = Ka^3
\end{equation*}

where \(K = 1.243 \times 10^{24}\text{,}\)\(a\) is measured in miles, and \(p\) is measured in years.

Solve Kepler’s law for \(p\) in terms of \(a\text{.}\)

Find the period of Mars if its average distance from the Sun is \(1.417 \times 10^8\) miles.

Answer.

\(\displaystyle p=K^{1/2}a^{3/2}\)

1.88 years

44.

Refer to Kepler’s law, \(p^2=Ka^3\text{,}\) in Problem 43.

Solve Kepler’s law for \(a\) in terms of \(p\text{.}\)

The period of Venus is 0.615 years. Find the distance from Venus to the Sun.