Chapter 10

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

Solve each equation. \(x^{2}-8 x=-12\)

Describe when Heron's formula might be useful.

Assume that all variables represent positive real numbers. $$ \sqrt{\frac{x^{2}}{4 y^{2}}} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{x} \cdot \sqrt[4]{x} \cdot \sqrt[8]{x^{3}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (5,1) and (8,5)

Factor each numerator and denominator. Then simplify if possible. $$ \frac{2 x-14}{2} $$

Find each power of \(i\). $$ i^{-6} $$

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

In your own words, explain why you think \(s\) in Heron's formula is called the semiperimeter.

Assume that all variables represent positive real numbers. $$ \sqrt{\frac{y^{10}}{9 x^{6}}} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[6]{y} \cdot \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (2,3) and (14,8)

Factor each numerator and denominator. Then simplify if possible. $$ \frac{8 x-24 y}{4} $$

Find each power of \(i\). $$ i^{-9} $$

The formula of the radius \(r\) of a sphere with surface area \(A\) is $$ r=\sqrt{\frac{A}{4 \pi}} $$ Rationalize the denominator of the radical expression in this formula.

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the function \(D(h)=111.7 \sqrt{h}\). Use this function for Exercises 79 and 80. Round your answers to two decimal places. Find the height that would allow a person to see 80 kilometers.

Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{z^{21}}{27 x^{3}}} $$

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

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (-3,2) and (1,-3)

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

Find each power of \(i\). $$ (2 i)^{6} $$

The formula for the radius \(r\) of a cone with height 7 centimeters and volume \(V\) is $$ r=\sqrt{\frac{3 V}{7 \pi}} $$ Rationalize the numerator of the radical expression in this formula.

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the function \(D(h)=111.7 \sqrt{h}\). Round your answers to two decimal places. Find the height that would allow a person to see 40 kilometers.

Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{64 a^{3}}{b^{9}}} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[5]{b^{2}}}{\sqrt[10]{b^{3}}} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (3,-2) and (-4,1)

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

Find each power of \(i\). $$ (5 i)^{4} $$

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

Explain why rationalizing the denominator does not change the value of the original expression.

$$ -\sqrt[3]{\frac{64 a^{3}}{b^{9}}} $$$$ \sqrt[4]{\frac{x^{4}}{16}} $$

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

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (-9,4) and (-8,1)

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

Find each power of \(i\). $$ (-3 i)^{5} $$

Explain why rationalizing the numerator does not change the value of the original expression.

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

$$ \sqrt[4]{\frac{x^{4}}{16}} $$$$ \sqrt[4]{\frac{y^{4}}{81 x^{4}}} $$

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

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (-5,-2) and (-6,-6)

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

Find each power of \(i\). $$ (-2 i)^{7} $$

Simplify. $$ \frac{\frac{z}{5}+\frac{1}{10}}{\frac{z}{20}-\frac{z}{5}} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(0) $$

Use rational expressions to write as a single radical expression. $$ \sqrt[5]{7} \cdot \sqrt[3]{y} $$

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

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

Simplify. $$ \frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y}-\frac{1}{x}} $$

Given \(\frac{\sqrt[3]{5 y}}{\sqrt[3]{4}}\), rationalize the denominator by following parts (a) and (b). a. Multiply the numerator and denominator by \(\sqrt[3]{16}\). b. Multiply the numerator and denominator by \(\sqrt[3]{2}\). c. What can you conclude from parts (a) and (b)?