Chapter 7

In \(3-37,\) express each power as a rational number in simplest form. $$ 27^{\frac{4}{3}} $$

Solve each equation and check. \(3^{x}=27\)

In \(3-17\) solve each equation and check. $$ x^{5}=3,125 $$

Simplify each expression. In each exercise, all variables are positive. \(x^{9} y^{7} \div\left(x^{8} y^{7}\right)\)

A bank offers certificates of deposit with variable compounding periods. a. Joe invested \(\$ 1,000\) at 6\(\%\) per year compounded yearly. Find the values of Joe's investment at the end of each of the first five years. b. Sue invested \(\$ 1,000\) at 6\(\%\) per year compounded monthly. Find the values of Sue's investment at the end of each of the first five years. c. Who had more money after the end of the fifth year? d. The annual percentage yield (APY) is the amount an investment actually increases during one year. Find the APY for Joe and Sue's certificates of deposit. Is the APY of each investment equal to 6\(\% ?\)

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ -2 x^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 10,000^{\frac{3}{4}} $$

Solve each equation and check. \(5^{x}=\frac{1}{5}\)

In \(3-17\) solve each equation and check. $$ z^{\frac{1}{2}}=\sqrt{81} $$

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

a. When Kyle was born, his grandparents invested \(\$ 5,000\) in a college fund that paid 4\(\%\) per year, compounded yearly. What was the value of this investment when Kyle was ready for college at age 18\(?\) (Note that \(r=0.04 . )\) b. If Kyle's grandparents had invested the \(\$ 5,000\) in a fund that paid 4\(\%\) compounded continuously, what would have been the value of the fund after 18 years?

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ (-2 x)^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 32^{\frac{4}{3}} $$

Solve each equation and check. \(7^{x}=\frac{1}{49}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ x^{-3}=24 $$

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

A trust fund of \(\$ 2.5\) million was donated to a charitable organization. Once each year the organization spends 2\(\%\) of the value of the fund so that the fund decreases by 2\(\% .\) What will be the value of the fund after 25 years?

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \left(\frac{3}{4}\right)^{0} $$

In \(3-37,\) express each power as a rational number in simplest form. $$ 9^{-\frac{1}{2}} $$

Solve each equation and check. \(4^{x+2}=4^{2 x}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ y^{\frac{2}{3}}=6 $$

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

The decay constant of a radioactive element is \(-0.533\) per minute. If a sample of the element weighs 50 grams, what will be its weight after 2 minutes?

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \frac{3^{0}}{4} $$

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

Solve each equation and check. \(3^{x+1}=3^{2 x+3}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ a^{-\frac{3}{4}}=0.85 $$

Simplify each expression. In each exercise, all variables are positive. \(\frac{3\left(x^{3}\right)^{4} y^{5}}{3 x^{7}}\)

The population of a small town decreased continually by 2\(\%\) each year. If the population of the town is now \(37,000,\) what will be the population 8 years from now if this trend continues?

In \(3-37,\) express each power as a rational number in simplest form. $$ 100^{-\frac{3}{2}} $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \frac{3^{0}}{4^{0}} $$

Solve each equation and check. \(6^{3 x}=6^{x-1}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ 3 z^{3}+2=27 $$

Simplify each expression. In each exercise, all variables are positive. \(\frac{-x^{4} y^{6}}{\left(-x^{3} y^{4}\right)}\)

A piece of property was valued at \(\$ 50,000\) at the end of \(1990 .\) Property values in the city where this land is located increase by 10\(\%\) each year. The value of the land increases continuously. What is the property worth at the end of 2010\(?\)

In \(3-37,\) express each power as a rational number in simplest form. $$ 125^{-\frac{3}{3}} $$

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \frac{3 x^{0}}{(4 x)^{0}} $$

Solve each equation and check. \(3^{x+2}=9^{x}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ 5+b^{5}=56 $$

Simplify each expression. In each exercise, all variables are positive. \(\left(\frac{x^{3} y^{5}}{\left(x y^{2}\right)^{2}}\right)^{2}\)

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=x^{-3} \cdot x^{4} ; \mathrm{f}(1) $$

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

Solve each equation and check. \(25^{x}=5^{x+3}\)

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ (3 w)^{9}+2=81 $$

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

The number of wolves in a wildlife preserve is estimated to have increased continually by 3\(\%\) per year. If the population is now estimated at \(5,400\) wolves, how many were present 10 years ago?

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=x+x^{-5} ; \mathrm{f}(3) $$

Solve each equation and check. \(49^{x}=7^{3 x+1}\)

Solve for \(x\) and check: \(\frac{x^{\frac{1}{3}}}{x^{\frac{2}{3}}}=10 .\) Use the rule for the division of powers with like bases to simplify the left side of the equation.

Simplify each expression. In each exercise, all variables are positive. \(\left(\frac{2 a^{3}}{a^{2}}\right)^{5} \cdot b\)