Chapter 7

Solve each equation. See Sections 2.1 and 5.8. $$ x^{3}=x $$

a. Add: \(2 \sqrt{5}+\sqrt{5}\) b. Multiply: \(2 \sqrt{5} \cdot \sqrt{5}\) c. Describe the differences in parts (a) and (b).

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}+1 $$

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

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

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-2,-1),(-8,6) $$

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. (GRAPH CANT COPY)

Multiply: \((\sqrt{2}+\sqrt{3}-1)^{2}\)

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}-2 $$

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

Simplify. \(\frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y}-\frac{1}{x}}\)

Find each power of \(i .\) See Example 6. $$ (5 i)^{4} $$

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-3,-4),(6,-8) $$

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. (GRAPH CANT COPY)

Multiply: \((\sqrt{5}-\sqrt{2}+1)^{2}\)

Identify the domain and then graph each function. \(g(x)=\sqrt[3]{x-1} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {g(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline 0 & {} \\ \hline 9 \\ \hline-7 \\ \hline \end{array} $$

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

Find the error in each solution and correct. \(\begin{array}{r} {\sqrt{5 x-1}+4=7} \\ {(\sqrt{5 x-1}+4)^{2}=7^{2}} \\ {5 x-1+16=49} \\ {5 x=34} \\ {x=\frac{34}{5}} \end{array}\)

Find each power of \(i .\) See Example 6. $$ (-3 i)^{5} $$

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (7,3),(-1,-3) $$

Given \(\frac{\sqrt{5 y^{3}}}{\sqrt{12 x^{3}}}\) rationalize the denominator by following parts (a) and (b). a. Multiply the numerator and denominator by \(\sqrt{12 x^{3}}\) b. Multiply the numerator and denominator by \(\sqrt{3 x}\) c. What can you conclude from parts (a) and (b)?

Explain how simplifying \(2 x+3 x\) is similar to simplifying \(2 \sqrt{x}+3 \sqrt{x}\)

Identify the domain and then graph each function. \(g(x)=\sqrt[3]{x+1} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {g(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline-2 & {} \\ \hline 7 \\ \hline-9 \\ \hline \end{array} $$

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

Find the error in each solution and correct. \(\sqrt{2 x+3}+4=1\) \(\begin{aligned} \sqrt{2 x+3} &=5 \\\\(\sqrt{2 x+3})^{2} &=5^{2} \\ 2 x+3 &=25 \\ 2 x &=22 \\ x &=11 \end{aligned}\)

Find each power of \(i .\) See Example 6. $$ (-2 i)^{7} $$

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (-2,5),(-1,6) $$

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)?

Explain how multiplying \((x-2)(x+3)\) is similar to multiplying \((\sqrt{x}-\sqrt{2})(\sqrt{x}+3)\)

Simplify each exponential expression. $$ \left(-2 x^{3} y^{2}\right)^{5} $$

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

Solve: \(\sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3}\)

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ \left(\frac{1}{2}, \frac{3}{8}\right),\left(-\frac{3}{2}, \frac{5}{8}\right) $$

Simplify each exponential expression. $$ \left(4 y^{6} z^{7}\right)^{3} $$

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

The cost \(C(x)\) in dollars per day to operate a small delivery service is given by \(C(x)=80 \sqrt[3]{x}+500,\) where \(x\) is the number of deliveries per day. In July, the manager decides that it is necessary to keep delivery costs below \(\$ 1620.00 .\) Find the greatest number of deliveries this company can make per day and still keep overhead below \(\$ 1620.00 .\)

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ \left(-\frac{2}{5}, \frac{7}{15}\right),\left(-\frac{2}{5},-\frac{4}{15}\right) $$

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section. $$ \frac{5}{\sqrt{27}} $$

Simplify each exponential expression. $$ \left(-3 x^{2} y^{3} z^{5}\right)\left(20 x^{5} y^{7}\right) $$

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

Consider the equations \(\sqrt{2 x}=4\) and \(\sqrt[3]{2 x}=4\) a. Explain the difference in solving these equations. b. Explain the similarity in solving these equations.

Use synthetic division to divide the following. See Section 6.4 $$ \left(x^{3}-6 x^{2}+3 x-4\right) \div(x-1) $$

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (\sqrt{2}, 3 \sqrt{5}),(\sqrt{2},-2 \sqrt{5}) $$

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section. When rationalizing the denominator of \(\frac{\sqrt{5}}{\sqrt{7}}\) explain why both the numerator and the denominator must be multiplied by \(\sqrt{7}\)

Simplify each exponential expression. $$ \left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right) $$

Use rational expressions to write as a single radical expression. $$ \sqrt[4]{5} \cdot \sqrt[3]{x} $$

Explain why proposed solutions of radical equations must be checked.

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (\sqrt{8},-\sqrt{12}),(3 \sqrt{2}, 7 \sqrt{3}) $$

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section. When rationalizing the numerator of \(\frac{\sqrt{5}}{\sqrt{7}}\) explain why both

Simplify each exponential expression. $$ \frac{7 x^{-1} y}{14\left(x^{5} y^{2}\right)^{-2}} $$