Writing to Learn Irrational Powers A negative number to an

In Exercises 1–6, write the following expressions using only positive integer powers. \(x^{2 / 3}\)

In Exercises 1–6, write the following expressions using only positive integer powers. \(p^{5 / 2}\)

In Exercises 1–6, write the following expressions using only positive integer powers. \(d^{-2}\)

In Exercises 1–6, write the following expressions using only positive integer powers. \(x^{-7}\)

In Exercises 1–6, write the following expressions using only positive integer powers. \(q^{-4 / 5}\)

In Exercises 1–6, write the following expressions using only positive integer powers. \(m^{-1.5}\)

In Exercises 7–10, write the following expressions in the form \(k \cdot x^{a}\) using a single rational number for the power a. \(\sqrt{9 x^{3}}\)

In Exercises 7–10, write the following expressions in the form \(k \cdot x^{a}\) using a single rational number for the power a. \(\sqrt[3]{8 x^{5}}\)

In Exercises 7–10, write the following expressions in the form \(k \cdot x^{a}\) using a single rational number for the power a. \(\sqrt[3]{\frac{5}{x^{4}}}\)

In Exercises 7–10, write the following expressions in the form \(k \cdot x^{a}\) using a single rational number for the power a. \(\frac{4 x}{\sqrt{32 x^{3}}}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(f(x)=-\frac{1}{2} x^{5}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(f(x)=9 x^{5 / 3}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(f(x)=3 \cdot 2^{x}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(f(x)=13\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(E(m)=m c^{2}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(K E(v)=\frac{1}{2} k v^{5}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(d=\frac{1}{2} g t^{2}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(V=\frac{4}{3} \pi r^{3}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(I=\frac{k}{d^{2}}\)

In Exercises 1–10, determine whether the function is a power function, given that c, g, k, and \(\pi\) represent constants. For those that are power functions, state the power and constant of variation. \(F(a)=m \cdot a\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=-4\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=3 x^{-5}\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(y=-6 x^{7}\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(y=-2 \cdot 5^{x}\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(S=4 \pi r^{2}\)

In Exercises 11–16, determine whether the function is a monomial function, given that l and \(\pi\) represent constants. For those that are monomial functions state the degree and leading coefficient. For those that are not, explain why not. \(A=l w\)

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. The area A of an equilateral triangle varies directly as the square of the length s of its sides.

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. The volume V of a circular cylinder with fixed height is proportional to the square of its radius r.

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. The current I in an electrical circuit is inversely proportional to the resistance R, with constant of variation V.

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. Charles’s Law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T.

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. The energy E produced in a nuclear reaction is proportional to the mass m, with the constant of variation being \(c^{2}\), the square of the speed of light.

In Exercises 17–22, write the statement as a power function equation. Use k for the constant of variation if one is not given. The speed p of a free-falling object that has been dropped from rest varies as the square root of the distance traveled d, with a constant of variation \(k=\sqrt{2 g}\).

In Exercises 23–26, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. w = mg, where w and m are the weight and mass of an object and g is the constant acceleration due to gravity.

In Exercises 23–26, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. \(C=\pi D\), where C and D are the circumference and diameter of a circle and \(\pi\) is the usual mathematical constant.

In Exercises 23–26, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. n = c/v, where n is the refractive index of a medium, v is the velocity of light in the medium, and c is the constant velocity of light in free space.

In Exercises 23–26, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. \(d=p^{2} /(2 g)\), where d is the distance traveled by a free-falling object dropped from rest, p is the speed of the object, and g is the constant acceleration due to gravity.

In Exercises 27–30, state the power and constant of variation for the function, graph it, and analyze it in the manner of Example 2 of this section. \(f(x)=2 x^{4}\)

In Exercises 27–30, state the power and constant of variation for the function, graph it, and analyze it in the manner of Example 2 of this section. \(f(x)=-3 x^{3}\)

In Exercises 27–30, state the power and constant of variation for the function, graph it, and analyze it in the manner of Example 2 of this section. \(f(x)=\frac{1}{2} \sqrt[4]{x}\)

In Exercises 27–30, state the power and constant of variation for the function, graph it, and analyze it in the manner of Example 2 of this section. \(f(x)=-2 x^{-3}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=\frac{2}{3} x^{4}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=5 x^{3}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=-1.5 x^{5}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=-2 x^{6}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=\frac{1}{4} x^{8}\)

In Exercises 31–36, describe how to obtain the graph of the given monomial function from the graph of \(g(x)=x^{n}\) with the same power n. State whether ƒ is even or odd. Sketch the graph by hand and support your answer with a grapher. \(f(x)=\frac{1}{8} x^{7}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=-\frac{2}{3} x^{4}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=\frac{1}{2} x^{-5}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=2 x^{1 / 4}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=-x^{5 / 3}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=-2 x^{-2}\)

In Exercises 37–42, match the equation to one of the curves labeled in the figure. \(f(x)=1.7 x^{2 / 3}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=3 x^{1 / 4}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=-4 x^{2 / 3}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=-2 x^{4 / 3}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=\frac{2}{5} x^{5 / 2}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=\frac{1}{2} x^{-3}\)

In Exercises 43–48, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Before using a grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=-x^{-4}\)

In Exercises 49 and 50, data are given for y as a power function of x. Write an equation for the power function, and state its power and constant of variation.

In Exercises 49 and 50, data are given for y as a power function of x. Write an equation for the power function, and state its power and constant of variation.

Boyle’s Law The volume of an enclosed gas (at a constant temperature) varies inversely as the pressure. If the pressure of a 3.46-L sample of neon gas at a temperature of 302 K is 0.926 atm, what would the volume be at a pressure of 1.452 atm if the temperature does not change?

Charles’s Law The volume of an enclosed gas (at a constant pressure) varies directly as the absolute temperature. If the pressure of a 3.46-L sample of neon gas at a temperature of 302 K is 0.926 atm, what would the volume be at a temperature of 338 K if the pressure does not change?

Diamond Refraction Diamonds have the extremely high refraction index of n = 2.42 on average over the range of visible light. Use the formula from Exercise 25 and the fact that \(c \approx 3.00 \times 10^8\mathrm{\ m}/\mathrm{\sec}\) to determine the speed of light through a diamond.

Windmill Power The power P (in watts) produced by a windmill is proportional to the cube of the wind speed v (in mph). If a wind of 10 mph generates 15 watts of power, how much power is generated by winds of 20, 40, and 80 mph? Make a table and explain the pattern.

Keeping Warm For mammals and other warm-blooded animals to stay warm requires quite a bit of energy. Temperature loss is related to surface area, which is related to body weight, and temperature gain is related to circulation, which is related to pulse rate. In the final analysis, scientists have concluded that the pulse rate r of mammals is a power function of their body weight w. (a) Draw a scatter plot of the data in Table 2.12. (b) Find the power regression model. (c) Superimpose the regression curve on the scatter plot. (d) Use the regression model to predict the pulse rate for a 450-kg horse. Is the result close to the 38 beats/min reported by A. J. Clark in 1927?

Even and Odd Functions If n is an integer, \(n \geq 1\), prove that \(f(x)=x^{n}\) is an odd function if n is odd and is an even function if n is even

Light Intensity Velma and Reggie gathered the data in Table 2.13 using a 100-watt light bulb and a Calculator-Based Laboratory™ (CBL™) with a light-intensity probe. (a) Draw a scatter plot of the data in Table 2.13 (b) Find the power regression model. Is the power close to the theoretical value of \(a=-2 ?\) (c) Superimpose the regression curve on the scatter plot. (d) Use the regression model to predict the light intensity at distances of 1.7 m and 3.4 m.

True or False The function \(f(x)=x^{-2 / 3}\) is even. Justify your answer.

True or False The graph \(f(x)=x^{1 / 3}\) is symmetric about the y-axis. Justify your answer.

In Exercises 60–63, solve the problem without using a calculator Multiple Choice Let \(f(x)=2 x^{-1 / 2}\). What is the value of ƒ(4)? (A) 1 (B) -1 (C) \(2 \sqrt{2}\) (D) \(\frac{1}{2 \sqrt{2}}\) (E) 4

In Exercises 60–63, solve the problem without using a calculator Multiple Choice Let \(f(x)=-3 x^{-1 / 3}\). Which of the following statements is true? (A) ƒ(0) = 0 (B) ƒ(-1) = -3 (C) ƒ(1) = 1 (D) ƒ(3) = 3 (E) ƒ(0) is undefined.

In Exercises 60–63, solve the problem without using a calculator Multiple Choice Let \(f(x)=x^{2 / 3}\). Which of the following statements is true? (A) ƒ is an odd function. (B) ƒ is an even function. (C) ƒ is neither an even nor an odd function. (D) The graph ƒ is symmetric with respect to the x-axis. (E) The graph ƒ is symmetric with respect to the origin

In Exercises 60–63, solve the problem without using a calculator Multiple Choice Which of the following is the domain of the function \(f(x)=x^{3 / 2} ?\) (A) All reals (B) \([0, \infty)\) (C) \((0, \infty)\) (D) \((-\infty, 0)\) (E) \((-\infty, 0) \cup(0, \infty)\)

Group Activity Rational Powers Working in a group of three students, investigate the behavior of power functions of the form \(f(x)=k \cdot x^{m / n}\), where m and n are positive with no factors in common. Have one group member investigate each of the following cases: • n is even • n is odd and m is even • n is odd and m is odd For each case, decide whether ƒ is even, ƒ is odd, or ƒ is undefined for \(x<0\). Solve graphically and confirm algebraically in a way to convince the rest of your group and your entire class.

Comparing the Graphs of Power Functions Graph the functions in the stated windows and explain how the graphs are alike and how they are different. Consider the relevant aspects of analysis from Example 2. Which ordered pairs do all four graphs have in common? (a) \(f(x)=x^{-1},\ g(x)=x^{-2},\ h(x)=x^{-3}\text{, and }k(x)=x^{-4}\text{ in }\text{ the windows }[0,\ 1]\text{ by }[0,\ 5],[0,\ 3]\text{ by }[0,\ 3]\text{, and }[-2,\ 2]\text{ by }[-2,\ 2].\) (b) \(f(x)=x^{1/2},\ g(x)=x^{1/3},\ h(x)=x^{1/4},\text{ and }k(x)=x^{1/5}\text{ in }\text{ the windows }[0,\ 1]\text{ by }[0,\ 1],\ [0,\ 3]\text{ by }[0,\ 2]\text{, and }[-3,\ 3]\text{ by }[-2,\ 2].\)

Writing to Learn Irrational Powers A negative number to an irrational power is undefined. Analyze the graphs of \(f(x)=x^{\pi},\ x^{1/\pi},\ x^{-\pi},\ -x^{\pi},\ -x^{1/\pi},\text{ and }-x^{-\pi}\). Prepare a sketch of all six graphs on one set of axes, labeling each of the curves. Write an explanation for why each graph is positioned and shaped as it is.

Planetary Motion Revisited Convert the time and distance units in Table 2.10 to the Earth-based units of years and astronomical units using 1 yr = 365.2 days and 1 AU = 149.6 Gm. Use this “re-expressed” data to obtain a power function model. Show algebraically that this model closely approximates Kepler’s equation \(T^{2}=a^{3}\).

Free Fall Revisited The speed p of an object is the absolute value of its velocity v. The distance traveled d by an object dropped from an initial height with a current height s is given by \(d=s_{0}-s\) until it hits the ground. Use this information and the free-fall motion formulas from Section 2.1 to prove that \(d=\frac{1}{2} g t^{2}, p=g t\), and therefore \(p=\sqrt{2 g d}\). Do the results of Example 6 approximate this last formula?

Prove that g(x) = 1/ƒ(x) is even if and only if ƒ(x) is even and that g(x) = 1/ƒ(x) is odd if and only if ƒ(x) is odd.

Use the results in Exercise 69 to prove that \(g(x)=x^{-a}\) is even if \(f(x)=x^{a}\) and only if is even and that is odd if and only if \(f(x)=x^{a}\) is odd.

Joint Variation If a variable z varies as the product of the variables x and y, we say z varies jointly as x and y, and we write \(z=k \cdot x \cdot y\), where k is the constant of variation. Write a sentence that expresses the relationship in each of the following formulas, using the language of joint variation. (a) \(F=m \cdot a\), where F and a are the force and acceleration acting on an object of mass m. (b) \(K E=(1 / 2) m \cdot v^{2}\), where KE and v are the kinetic energy and velocity of an object of mass m. (c) \(F=G \cdot m_{1} \cdot m_{2} / r^{2}\), where F is the force of gravity acting on objects of masses \(m_{1} \text { and } m_{2}\) with a distance r between their centers and G is the universal gravitational constant.