Chapter 7

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (-3,2) \text { and }(1,-3) $$

Rationalize each numerator. See Example 7. $$ \frac{\sqrt{2}-1}{\sqrt{2}+1} $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{7 x-7 y}{x^{2}-y^{2}} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ f(-1) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[9]{y^{6} z^{3}} $$

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the func\(\operatorname{tion} D(h)=111.7 \sqrt{h} .\) Find the height that would allow a person to see 40 kilometers.

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

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (3,-2) \text { and }(-4,1) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{x^{3}-8}{4 x-8} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(-19) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[12]{a^{8} b^{4}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (-9,4) \text { and }(-8,1) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{6 a^{2} b-9 a b}{3 a b} $$

Find each power of \(i .\) See Example 6. $$ 0.81 . i^{8} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ f(3) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[10]{a^{5} b^{5}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (-5,-2) \text { and }(-6,-6) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{14 r-28 r^{2} s^{2}}{7 r s} $$

Find each power of \(i .\) See Example 6. $$ i^{10} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ f(2) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{(x+3)^{2}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (0,-\sqrt{2}) \text { and }(\sqrt{3}, 0) $$

Solve each equation. See Sections 2.1 and 5.8. $$ 2 x-7=3(x-4) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{-4+2 \sqrt{3}}{6} $$

Find each power of \(i .\) See Example 6. $$ i^{21} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ g(1) $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[8]{(y+1)^{4}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (-\sqrt{5}, 0) \text { and }(0, \sqrt{7}) $$

Solve each equation. See Sections 2.1 and 5.8. $$ 9 x-4=7(x-2) $$

Factor each mumerator and denominator. Then simplify if possible. $$ \frac{-5+10 \sqrt{7}}{5} $$

Identify the domain and then graph each function. $$ f(x)=\sqrt{x}+2 $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (1.7,-3.6) \text { and }(-8.6,5.7) $$

Solve each equation. See Sections 2.1 and 5.8. $$ (x-6)(2 x+1)=0 $$

Identify the domain and then graph each function. $$ f(x)=\sqrt{x}-2 $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y^{2}} \cdot \sqrt[6]{y} $$

Find each power of \(i .\) See Example 6. $$ i^{40} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (9.6,2.5) \text { and }(-1.9,-3.7) $$

Solve each equation. See Sections 2.1 and 5.8. $$ (y+2)(5 y+4)=0 $$

Find the area and perimeter of the trapezoid. (Hint: The area of a trapezoid is the product of half the height \(6 \sqrt{3}\) meters and the sum of the bases \(2 \sqrt{63}\) and \(7 \sqrt{7}\) meters.)

Identify the domain and then graph each function. \(f(x)=\sqrt{x-3} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {f(x)} \\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 7 \\ \hline 12 & {} \\ \hline \end{array} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}} $$

Simplify. \(\frac{\frac{x}{6}}{\frac{2 x}{3}+\frac{1}{2}}\)

Solve each equation. See Sections 2.1 and 5.8. $$ x^{2}-8 x=-12 $$

a. Add: \(\sqrt{3}+\sqrt{3}\) b. Multiply: \(\sqrt{3} \cdot \sqrt{3}\) c. Describe the differences in parts (a) and (b).

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (6,-8),(2,4) $$

Identify the domain and then graph each function. \(f(x)=\sqrt{x+1} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {f(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 3 & {} \\ \hline 8 & {} \\ \hline \end{array} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[4]{a}}{\sqrt[5]{a}} $$

Simplify. \(\frac{\frac{1}{y}+\frac{4}{5}}{-\frac{3}{20}}\)

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (3,9),(7,11) $$