Chapter 7

Write each number as a power. 32

In \(3-17\) solve each equation and check. $$ b^{-5}=\frac{1}{32} $$

a. Sketch the graph of \(f(x)=2^{x} .\) b. Sketch the graph of the image of \(f(x)=2^{x}\) under a reflection in the \(x\) -axis. c. Write an equation for the function whose graph was sketched in part b.

Simplify each expression. In each exercise, all variables are positive. \(\left(2 y^{4}\right)^{3}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{3} e^{5} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ \frac{3^{0}}{4^{-2}} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 32^{\frac{1}{5}} $$

Write each number as a power. \(\frac{1}{8}\)

In \(3-17\) solve each equation and check. $$ 2 y^{-1}=12 $$

a. Sketch the graph of \(f(x)=1.2^{x} .\) b. Sketch the graph of the image of \(f(x)=1.2^{x}\) under a reflection in the \(x\) -axis. c. Write an equation for the function whose graph was sketched in part b.

Simplify each expression. In each exercise, all variables are positive. \(10^{2} \cdot 10^{4}\)

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{3}+e^{5} $$

In \(3-10,\) write each expression as a rational number without an exponent. $$ \frac{(2 \cdot 5)^{-4}}{5^{-2}} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ (3 \times 12)^{\frac{1}{2}} $$

Write each number as a power. \(\frac{1}{216}\)

In \(3-17\) solve each equation and check. $$ 9 a^{-\frac{3}{4}}=\frac{1}{3} $$

a. Make a table of values for \(e^{x}\) for integral values of \(x\) from \(-2\) to 3 b. Sketch the graph of \(\mathrm{f}(x)=e^{x}\) by plotting points and joining them with a smooth curve: c. From the graph, estimate the value of \(e^{\frac{1}{2}}\) and compare your answer to the value given by a calculator.

Simplify each expression. In each exercise, all variables are positive. \(-2^{6} \cdot 2^{2}\)

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A_{0}=50, r=2 \%, n=12, t=1 $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ 7^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ (2 \times 8)^{\frac{1}{4}} $$

Write each number as a power. 0.001

The population of the United States can be modeled by the function \(\mathrm{p}(x)=80.21 e^{0.131 x}\) where \(x\) is the number of decades (ten year periods) since 1900 and \(\mathrm{p}(x)\) is the population in millions. a. Graph \(\mathrm{p}(x)\) over the interval \(0 \leq x \leq 15 .\) b. If the population of the United States continues to grow at this rate, predict the population in the years 2010 and \(2020 .\)

In \(3-17\) solve each equation and check. $$ 5 x^{\frac{3}{4}}=40 $$

Simplify each expression. In each exercise, all variables are positive. \(x^{4} \cdot x^{2} y^{3}\)

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=400, r=5 \%, n=4, t=3 $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ (-5)^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 5(81)^{\frac{1}{4}} $$

In \(1986,\) the worst nuclear power plant accident in history occurred in the Chernobyl Nuclear Power Plant located in the Ukraine. On April \(26,\) one of the reactors exploded, releasing large amounts of radioactive isotopes into the atmosphere. The amount of plutonium present after \(t\) years can be modeled by the function: $$ y=P e^{-0.0000288 t} $$ where \(P\) represents the amount of plutonium that is released. a. Graph this function over the interval \(0 \leq t \leq 100,000\) and \(P=10\) grams. b. If 10 grams of the isotope plutonium- 239 were released into the air, to the nearest hundredth, how many grams will be left after 10 years? After 100 years? c. Using the graph, approximate how long it will take for the 10 grams of plutonium- 239 to decay to 1 gram.

Write each number as a power. 0.125

In \(3-17\) solve each equation and check. $$ 5 x^{\frac{1}{2}}+7=22 $$

Simplify each expression. In each exercise, all variables are positive. \(x y^{5} \cdot x y^{2}\)

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=100, A_{0}=25, n=1, t=2 $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ x^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ -4(1,000)^{\frac{1}{3}} $$

a. Graph the functions \(y=x^{4}\) and \(y=4^{x}\) on a graphing calculator using the following viewing windows: (1) \(\operatorname{Xmin}=0, \operatorname{Xmax}=3,\) Ymin \(=0,\) Ymax \(=50\) (2) \(X \min =0, \operatorname{Xmax}=5, Y \min =0,\) Ymax \(=500\) (3) \(X \min =0, \operatorname{Xmax}=5, Y \min =0, Y \max =1,000\) b. How many points of intersection can you find? Find the coordinates of these intersection points to the nearest tenth. c. Which function grows more rapidly for increasing values of \(x ?\)

In \(3-17\) solve each equation and check. $$ 14-4 b^{\frac{1}{3}}=2 $$

Simplify each expression. In each exercise, all variables are positive. \(-\left(3 x^{3}\right)^{2}\)

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=25, A_{0}=200, r=-50 \%, n=1 $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ -4^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 49^{\frac{3}{2}} $$

In \(3-17\) solve each equation and check. $$ (2 x)^{\frac{1}{2}}+3=15 $$

Simplify each expression. In each exercise, all variables are positive. \(\left(-3 x^{3}\right)^{2}\)

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ (4 x)^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 8^{\frac{5}{3}} $$

Solve each equation and check. \(2^{x}=16\)

In \(3-17\) solve each equation and check. $$ 3 a^{3}=81 $$

Simplify each expression. In each exercise, all variables are positive. \(x^{8} y^{6} \div\left(x^{3} y^{5}\right)\)

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=6, A_{0}=36, n=1, t=4 $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ 4 x^{0} $$