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 \(\dfrac{3}{2}= 3\left(\dfrac{1}{2}\right)\text{,}\) and thus by the third law of exponents, we can write
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.
\(\qquad\qquad 81\)^\(3\)÷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.
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.
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.
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.
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.
\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*}
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*}
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:
“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.
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 \(P=\dfrac{F}{A}\text{.}\)
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 \(d\text{?}\)
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.
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
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
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)
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?
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.
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.
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.
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)
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?
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
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.)
