Chapter 10

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{y^{11 / 3}}{\left(y^{5}\right)^{1 / 3}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (8 \sqrt{y}+z)(4 \sqrt{y}-1) $$

Simplify. Assume that all variables represent positive real numbers. $$ -\sqrt{20 a b^{6}} $$

Multiply. Write your answers in the form \(a+b i\). $$ (2-2 i)^{2} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{2 x}}{11}\)

Simplify. Assume that the variables represent any real number. $$ \sqrt[3]{x^{3}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(3 x^{1 / 4}\right)^{3}}{x^{1 / 12}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{a}-4)(\sqrt[3]{a}+5) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt{9 x^{7} y^{9}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{4}{i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{y}}{7}\)

Simplify. Assume that the variables represent any real number. $$ \sqrt[5]{x^{5}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(2 x^{1 / 5}\right)^{4}}{x^{3 / 10}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{a}+2)(\sqrt[3]{a}+7) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt{12 r^{9} s^{12}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{5}{6 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{7}{8}}\)

Simplify. Assume that the variables represent any real number. $$ \sqrt{(x-5)^{2}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(y^{3} z\right)^{1 / 6}}{y^{-1 / 2} z^{1 / 3}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ 6(\sqrt{2}-2) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{125 r^{9} s^{12}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{7}{4+3 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{25}{2}}\)

Simplify. Assume that the variables represent any real number. $$ \sqrt{(y-6)^{2}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(m^{2} n\right)^{1 / 4}}{m^{-1 / 2} n^{5 / 8}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{5}(6-\sqrt{5}) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{8 a^{6} b^{9}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{9}{1-2 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{3 x^{5}}}{10}\)

Solve. Give exact answers and two-decimal-place approximations where appropriate. A wire is needed to support a vertical pole 15 feet tall. The cable will be anchored to a stake 8 feet from the base of the pole. How much cable is needed?

Simplify. Assume that the variables represent any real number. $$ \sqrt{x^{2}+4 x+4} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3} y^{2}\right)^{1 / 4}}{\left(x^{-5} y^{-1}\right)^{-1 / 2}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{2}(\sqrt{2}+x \sqrt{6}) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{32 x^{12} y^{5}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{6 i}{1-2 i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{9 y}{7}}\)

Solve. Give exact answers and two-decimal-place approximations where appropriate. The tallest structure in the United States is a TV tower in Blanchard, North Dakota. Its height is 2063 feet. A 2382 -foot length of wire is to be used as a guy wire attached to the top of the tower. Approximate to the nearest foot how far from the base of the tower the guy wire must be anchored. (Source: U.S. Geological Survey)

Simplify. Assume that the variables represent any real number. $$ \sqrt{x^{2}-8 x+16} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(a^{-2} b^{3}\right)^{1 / 8}}{\left(a^{-3} b\right)^{-1 / 4}} $$

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{3}(\sqrt{3}-2 \sqrt{5 x}) $$

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{162 x^{7} y^{20}} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{3 i}{5+i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{18 x^{4} y^{6}}{3 z}}\)

A spotlight is mounted on the eaves of a house, 12 feet above the ground. A flower bed runs between the house and the sidewalk, so the closest the ladder can be placed to the house is 5 feet. How long a ladder is needed so that an electrician can reach the place where the light is mounted?

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[6]{x^{3}} $$

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{14}}{\sqrt{7}}\)

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (2 \sqrt{7}+3 \sqrt{5})(\sqrt{7}-2 \sqrt{5}) $$

Assume that all variables represent positive real numbers. $$ -\sqrt{121} $$

Divide. Write your answers in the form \(a+b i\) $$ \frac{3+5 i}{1+i} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{8 x^{5} y}{2 z}}\)