Chapter 7

Multiply and then simplify if possible. $$ \sqrt{3 x}(\sqrt{3}-\sqrt{x}) $$

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. $$ \frac{a^{1 / 4} a^{-1 / 2}}{a^{2 / 3}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (-2-4 i)-(6-8 i) $$

Simplify. See Examples 3 and 4 $$ \sqrt[3]{40 y^{10}} $$

Multiply and then simplify if possible. $$\sqrt{5 y}(\sqrt{y}+\sqrt{5})$$

Simplify. Assume that the variables represent any real number. $$\sqrt{x^{2}+4 x+4}$$ (Hint: Factor the polynomial first.)

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3}\right)^{1 / 2}}{x^{7 / 2}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ -3 i(-1+9 i) $$

Simplify. See Examples 3 and 4 $$ -\sqrt{32 a^{8} b^{7}} $$

Rationalize each denominator. See Example 4. $$ \frac{2 \sqrt{3}+\sqrt{6}}{4 \sqrt{3}-\sqrt{6}} $$

Multiply and then simplify if possible. $$(2 \sqrt{x}-5)(3 \sqrt{x}+1)$$

Simplify. Assume that the variables represent any real number. $$\sqrt{x^{2}-8 x+16}$$ (Hint: Factor the polynomial first.)

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

$$ -5 i(-2+i) $$

Rationalize each denominator. See Example 4. $$ \frac{4 \sqrt{5}+\sqrt{2}}{2 \sqrt{5}-\sqrt{2}} $$

Multiply and then simplify if possible. $$ (8 \sqrt{y}+z)(4 \sqrt{y}-1) $$

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

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

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{4-5 i}{2 i} $$

Simplify. See Examples 3 and 4 $$ \sqrt{9 x^{7} y^{9}} $$

Multiply and then simplify if possible. $$ (\sqrt[3]{a}-4)(\sqrt[3]{a}+5) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{125} $$

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

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{6+8 i}{3 i} $$

Multiply and then simplify if possible. $$ (\sqrt[3]{a}+2)(\sqrt[3]{a}+7) $$

Simplify. See Examples 3 and 4 $$ \sqrt{12 r^{9} s^{12}} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[3]{8 x^{3}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(y^{3} z\right)^{16}}{y^{-1 / 2} z^{1 / 3}} $$

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (4+i)(5+2 i) $$

Simplify. See Examples 3 and 4 $$ \sqrt[3]{125 r^{9} s^{12}} $$

Multiply and then simplify if possible. $$ 6(\sqrt{2}-2) $$

Rationalize each numerator. See Examples 5 and 6. $$ \sqrt{\frac{18}{5}} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{16 x^{8}} $$

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

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (3+i)(2+4 i) $$

Multiply and then simplify if possible. $$ \sqrt{5}(6-\sqrt{5}) $$

Simplify. See Examples 3 and 4 $$ \sqrt[3]{8 a^{6} b^{9}} $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{y^{12}} $$

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

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?

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (6-2 i)(3+i) $$

Multiply and then simplify if possible. $$ \sqrt{2}(\sqrt{2}+x \sqrt{6}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[3]{y^{12}} $$

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

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.

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (2-4 i)(2-i) $$

Multiply and then simplify if possible. $$ \sqrt{3}(\sqrt{3}-2 \sqrt{5 x}) $$

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{25 a^{2} b^{20}} $$

Multiply. $$ y^{1 / 2}\left(y^{1 / 2}-y^{2 / 3}\right) $$