# Chapter 8 Review Exercises

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## Chapter Review Exercises

### Simplify Expressions with Roots

##### Exercise $$\PageIndex{1}$$ Simplify Expressions with Roots

In the following exercises, simplify.

1. $$\sqrt{225}$$
2. $$-\sqrt{16}$$
1. $$-\sqrt{169}$$
2. $$\sqrt{-8}$$
1. $$\sqrt[3]{8}$$
2. $$\sqrt[4]{81}$$
3. $$\sqrt[5]{243}$$
1. $$\sqrt[3]{-512}$$
2. $$\sqrt[4]{-81}$$
3. $$\sqrt[5]{-1}$$

1.

1. $$15$$
2. $$-4$$

3.

1. $$2$$
2. $$3$$
3. $$3$$
##### Exercise $$\PageIndex{2}$$ Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

1. $$\sqrt{68}$$
2. $$\sqrt[3]{84}$$

1.

1. $$8<\sqrt{68}<9$$
2. $$4<\sqrt[3]{84}<5$$
##### Exercise $$\PageIndex{3}$$ Estimate and Approximate Roots

In the following exercises, approximate each root and round to two decimal places.

1. $$\sqrt{37}$$
2. $$\sqrt[3]{84}$$
3. $$\sqrt[4]{125}$$

1. Solve for yourself

##### Exercise $$\PageIndex{4}$$ Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

1. $$\sqrt[3]{a^{3}}$$
2. $$\sqrt[7]{b^{7}}$$
1. $$\sqrt{a^{14}}$$
2. $$\sqrt{w^{24}}$$
1. $$\sqrt[4]{m^{8}}$$
2. $$\sqrt[5]{n^{20}}$$
1. $$\sqrt{121 m^{20}}$$
2. $$-\sqrt{64 a^{2}}$$
1. $$\sqrt[3]{216 a^{6}}$$
2. $$\sqrt[5]{32 b^{20}}$$
1. $$\sqrt{144 x^{2} y^{2}}$$
2. $$\sqrt{169 w^{8} y^{10}}$$
3. $$\sqrt[3]{8 a^{51} b^{6}}$$

1.

1. $$a$$
2. $$|b|$$

3.

1. $$m^{2}$$
2. $$n^{4}$$

5.

1. $$6a^{2}$$
2. $$2b^{4}$$

##### Exercise $$\PageIndex{5}$$ Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

1. $$\sqrt{125}$$
2. $$\sqrt{675}$$
1. $$\sqrt[3]{625}$$
2. $$\sqrt[6]{128}$$

1. $$5\sqrt{5}$$

3.

1. $$5 \sqrt[3]{5}$$
2. $$2 \sqrt[6]{2}$$
##### Exercise $$\PageIndex{6}$$ Use the Product Property to Simplify Radical Expressions

In the following exercises, simplify using absolute value signs as needed.

1. $$\sqrt{a^{23}}$$
2. $$\sqrt[3]{b^{8}}$$
3. $$\sqrt[8]{c^{13}}$$
1. $$\sqrt{80 s^{15}}$$
2. $$\sqrt[5]{96 a^{7}}$$
3. $$\sqrt[6]{128 b^{7}}$$
1. $$\sqrt{96 r^{3} s^{3}}$$
2. $$\sqrt[3]{80 x^{7} y^{6}}$$
3. $$\sqrt[4]{80 x^{8} y^{9}}$$
1. $$\sqrt[5]{-32}$$
2. $$\sqrt[8]{-1}$$
1. $$8+\sqrt{96}$$
2. $$\frac{2+\sqrt{40}}{2}$$

2.

1. $$4\left|s^{7}\right| \sqrt{5 s}$$
2. $$2 a \sqrt[5]{3 a^{2}}$$
3. $$2|b| \sqrt[6]{2 b}$$

4.

1. $$-2$$
2. not real
##### Exercise $$\PageIndex{7}$$ Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

1. $$\sqrt{\frac{72}{98}}$$
2. $$\sqrt[3]{\frac{24}{81}}$$
3. $$\sqrt[4]{\frac{6}{96}}$$
1. $$\sqrt{\frac{y^{4}}{y^{8}}}$$
2. $$\sqrt[5]{\frac{u^{21}}{u^{11}}}$$
3. $$\sqrt[6]{\frac{v^{30}}{v^{12}}}$$
1. $$\sqrt{\frac{300 m^{5}}{64}}$$
1. $$\sqrt{\frac{28 p^{7}}{q^{2}}}$$
2. $$\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$$
3. $$\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$$
1. $$\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$$
2. $$\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$$
3. $$\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$$
1. $$\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$$
2. $$\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}$$
3. $$\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$$

1.

1. $$\frac{6}{7}$$
2. $$\frac{2}{3}$$
3. $$\frac{1}{2}$$

3. $$\frac{10 m^{2} \sqrt{3 m}}{8}$$

5.

1. $$\frac{1}{2|p q|}$$
2. $$\frac{2 c d \sqrt[5]{2 d^{2}}}{5}$$
3. $$\frac{|m n| \sqrt[6]{2}}{2}$$

### Simplify Rational Exponents

##### Exercise $$\PageIndex{8}$$ Simplify Expressions with $$a^{\frac{1}{n}}$$

In the following exercises, write as a radical expression.

1. $$r^{\frac{1}{2}}$$
2. $$s^{\frac{1}{3}}$$
3. $$t^{\frac{1}{4}}$$

1.

1. $$\sqrt{r}$$
2. $$\sqrt[3]{s}$$
3. $$\sqrt[4]{t}$$
##### Exercise $$\PageIndex{9}$$ Simplify Expressions with $$a^{\frac{1}{n}}$$

In the following exercises, write with a rational exponent.

1. $$\sqrt{21p}$$
2. $$\sqrt[4]{8q}$$
3. $$4\sqrt[6]{36r}$$

1. Solve for yourself

##### Exercise $$\PageIndex{10}$$ Simplify Expressions with $$a^{\frac{1}{n}}$$

In the following exercises, simplify.

1. $$625^{\frac{1}{4}}$$
2. $$243^{\frac{1}{5}}$$
3. $$32^{\frac{1}{5}}$$
1. $$(-1,000)^{\frac{1}{3}}$$
2. $$-1,000^{\frac{1}{3}}$$
3. $$(1,000)^{-\frac{1}{3}}$$
1. $$(-32)^{\frac{1}{5}}$$
2. $$(243)^{-\frac{1}{5}}$$
3. $$-125^{\frac{1}{3}}$$

1.

1. $$5$$
2. $$3$$
3. $$2$$

3.

1. $$-2$$
2. $$\frac{1}{3}$$
3. $$-5$$
##### Exercise $$\PageIndex{11}$$ Simplify Expressions with $$a^{\frac{m}{n}}$$

In the following exercises, write with a rational exponent.

1. $$\sqrt[4]{r^{7}}$$
2. $$(\sqrt[5]{2 p q})^{3}$$
3. $$\sqrt[4]{\left(\frac{12 m}{7 n}\right)^{3}}$$

1. Solve for yourself

##### Exercise $$\PageIndex{12}$$ Simplify Expressions with $$a^{\frac{m}{n}}$$

In the following exercises, simplify.

1. $$25^{\frac{3}{2}}$$
2. $$9^{-\frac{3}{2}}$$
3. $$(-64)^{\frac{2}{3}}$$
1. $$-64^{\frac{3}{2}}$$
2. $$-64^{-\frac{3}{2}}$$
3. $$(-64)^{\frac{3}{2}}$$

1.

1. $$125$$
2. $$\frac{1}{27}$$
3. $$16$$
##### Exercise $$\PageIndex{13}$$ Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

1. $$6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$$
2. $$\left(b^{15}\right)^{\frac{3}{5}}$$
3. $$\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$$
1. $$\frac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}$$
2. $$\left(\frac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}$$

1.

1. $$6^{3}$$
2. $$b^{9}$$
3. $$\frac{1}{w}$$

##### Exercise $$\PageIndex{14}$$ add and Subtract Radical Expressions

In the following exercises, simplify.

1. $$7 \sqrt{2}-3 \sqrt{2}$$
2. $$7 \sqrt[3]{p}+2 \sqrt[3]{p}$$
3. $$5 \sqrt[3]{x}-3 \sqrt[3]{x}$$
1. $$\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b}$$
2. $$8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}$$
1. $$\sqrt{48}+\sqrt{27}$$
2. $$\sqrt[3]{54}+\sqrt[3]{128}$$
3. $$6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}$$
1. $$\sqrt{80 c^{7}}-\sqrt{20 c^{7}}$$
2. $$2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}$$
1. $$3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}$$

1.

1. $$4\sqrt{2}$$
2. $$9\sqrt[3]{p}$$
3. $$2\sqrt[3]{x}$$

3.

1. $$7\sqrt{3}$$
2. $$7\sqrt[3]{2}$$
3. $$3\sqrt[4]{5}$$

5. $$37 y \sqrt{3}$$

##### Exercise $$\PageIndex{15}$$ Multiply Radical Expressions

In the following exercises, simplify.

1. $$(5 \sqrt{6})(-\sqrt{12})$$
2. $$(-2 \sqrt[4]{18})(-\sqrt[4]{9})$$
1. $$\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)$$
2. $$\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)$$

2.

1. $$126 x^{2} \sqrt{2}$$
2. $$48 a \sqrt[3]{a^{2}}$$
##### Exercise $$\PageIndex{16}$$ Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

1. $$\sqrt{11}(8+4 \sqrt{11})$$
2. $$\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})$$
1. $$(3-2 \sqrt{7})(5-4 \sqrt{7})$$
2. $$(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$$
1. $$(2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})$$
1. $$(4+\sqrt{11})^{2}$$
2. $$(3-2 \sqrt{5})^{2}$$
2. $$(7+\sqrt{10})(7-\sqrt{10})$$
3. $$(\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)$$

2.

1. $$71-22 \sqrt{7}$$
2. $$\sqrt[3]{x^{2}}-8 \sqrt[3]{x}+15$$

4.

1. $$27+8 \sqrt{11}$$
2. $$29-12 \sqrt{5}$$

6. $$\sqrt[3]{9 x^{2}}-4$$

##### Exercise $$\PageIndex{17}$$ Divide Square Roots

In the following exercises, simplify.

1. $$\frac{\sqrt{48}}{\sqrt{75}}$$
2. $$\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$$
1. $$\frac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}$$
2. $$\frac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}$$

2.

1. $$\frac{8 m^{4}}{3 n^{4}}$$
2. $$-\frac{x^{2}}{2 y^{2}}$$
##### Exercise $$\PageIndex{18}$$ rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

1. $$\frac{8}{\sqrt{3}}$$
2. $$\sqrt{\frac{7}{40}}$$
3. $$\frac{8}{\sqrt{2 y}}$$
1. $$\frac{1}{\sqrt[3]{11}}$$
2. $$\sqrt[3]{\frac{7}{54}}$$
3. $$\frac{3}{\sqrt[3]{3 x^{2}}}$$
1. $$\frac{1}{\sqrt[4]{4}}$$
2. $$\sqrt[4]{\frac{9}{32}}$$
3. $$\frac{6}{\sqrt[4]{9 x^{3}}}$$

2.

1. $$\frac{\sqrt[3]{121}}{11}$$
2. $$\frac{\sqrt[3]{28}}{6}$$
3. $$\frac{\sqrt[3]{9 x}}{x}$$
##### Exercise $$\PageIndex{19}$$ Rationalize a Two Term Denominator

In the following exercises, simplify.

1. $$\frac{7}{2-\sqrt{6}}$$
2. $$\frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}$$
3. $$\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$$

1. $$-\frac{7(2+\sqrt{6})}{2}$$

3. $$\frac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}$$

##### Exercise $$\PageIndex{20}$$ Solve Radical Equations

In the following exercises, solve.

1. $$\sqrt{4 x-3}=7$$
2. $$\sqrt{5 x+1}=-3$$
3. $$\sqrt[3]{4 x-1}=3$$
4. $$\sqrt{u-3}+3=u$$
5. $$\sqrt[3]{4 x+5}-2=-5$$
6. $$(8 x+5)^{\frac{1}{3}}+2=-1$$
7. $$\sqrt{y+4}-y+2=0$$
8. $$2 \sqrt{8 r+1}-8=2$$

2. no solution

4. $$u=3, u=4$$

6. $$x=-4$$

8. $$r=3$$

##### Exercise $$\PageIndex{21}$$ Solve Radical Equations with Two Radicals

In the following exercises, solve.

1. $$\sqrt{10+2 c}=\sqrt{4 c+16}$$
2. $$\sqrt[3]{2 x^{2}+9 x-18}=\sqrt[3]{x^{2}+3 x-2}$$
3. $$\sqrt{r}+6=\sqrt{r+8}$$
4. $$\sqrt{x+1}-\sqrt{x-2}=1$$

2. $$x=-8, x=2$$

4. $$x=3$$

##### Exercise $$\PageIndex{22}$$ Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

1. Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of $$75$$ square feet. Use the formula $$s=\sqrt{A}$$ to find the length of each side of his garden. Round your answers to th nearest tenth of a foot.
2. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $$175$$ feet. Use the formula $$s=\sqrt{24d}$$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

2. $$64.8$$ feet

##### Exercise $$\PageIndex{23}$$ Evaluate a Radical Function

In the following exercises, evaluate each function.

1. $$g(x)=\sqrt{6 x+1}$$, find
1. $$g(4)$$
2. $$g(8)$$
2. $$G(x)=\sqrt{5 x-1}$$, find
1. $$G(5)$$
2. $$G(2)$$
3. $$h(x)=\sqrt[3]{x^{2}-4}$$, find
1. $$h(-2)$$
2. $$h(6)$$
4. For the function $$g(x)=\sqrt[4]{4-4 x}$$, find
1. $$g(1)$$
2. $$g(-3)$$

2.

1. $$G(5)=2 \sqrt{6}$$
2. $$G(2)=3$$

4.

1. $$g(1)=0$$
2. $$g(-3)=2$$
##### Exercise $$\PageIndex{24}$$ Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

1. $$g(x)=\sqrt{2-3 x}$$
2. $$F(x)=\sqrt{\frac{x+3}{x-2}}$$
3. $$f(x)=\sqrt[3]{4 x^{2}-16}$$
4. $$F(x)=\sqrt[4]{10-7 x}$$

2. $$(2, \infty)$$

4. $$\left[\frac{7}{10}, \infty\right)$$

##### Exercise $$\PageIndex{25}$$ graph Radical Functions

In the following exercises,

1. find the domain of the function
2. graph the function
3. use the graph to determine the range
1. $$g(x)=\sqrt{x+4}$$
2. $$g(x)=2 \sqrt{x}$$
3. $$f(x)=\sqrt[3]{x-1}$$
4. $$f(x)=\sqrt[3]{x}+3$$

2.

1. domain: $$[0, \infty)$$

2. Figure 8.E.1
3. range: $$[0, \infty)$$

4.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.E.2
3. range: $$(-\infty, \infty)$$

### Use the Complex Number System

##### Exercise $$\PageIndex{26}$$ evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of $$i$$ and simplify if possible.

1. $$\sqrt{-100}$$
2. $$\sqrt{-13}$$
3. $$\sqrt{-45}$$

Solve for yourself

##### Exercise $$\PageIndex{27}$$ Add or Subtract Complex Numbers

In the following exercises, add or subtract.

1. $$\sqrt{-50}+\sqrt{-18}$$
2. $$(8-i)+(6+3 i)$$
3. $$(6+i)-(-2-4 i)$$
4. $$(-7-\sqrt{-50})-(-32-\sqrt{-18})$$

1. $$8 \sqrt{2} i$$

3. $$8+5 i$$

##### Exercise $$\PageIndex{28}$$ Multiply Complex Numbers

In the following exercises, multiply.

1. $$(-2-5 i)(-4+3 i)$$
2. $$-6 i(-3-2 i)$$
3. $$\sqrt{-4} \cdot \sqrt{-16}$$
4. $$(5-\sqrt{-12})(-3+\sqrt{-75})$$

1. $$23+14 i$$

3. $$-6$$

##### Exercise $$\PageIndex{29}$$ Multiply Complex Numbers

In the following exercises, multiply using the Product of Binomial Squares Pattern.

1. $$(-2-3 i)^{2}$$

1. $$-5-12 i$$

##### Exercise $$\PageIndex{30}$$ Multiply Complex Numbers

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

1. $$(9-2 i)(9+2 i)$$

Solve for yourself

##### Exercise $$\PageIndex{31}$$ divide Complex Numbers

In the following exercises, divide.

1. $$\frac{2+i}{3-4 i}$$
2. $$\frac{-4}{3-2 i}$$

1. $$\frac{2}{25}+\frac{11}{25} i$$

##### Exercise $$\PageIndex{32}$$ Simplify Powers of $$i$$

In the following exercises, simplify.

1. $$i^{48}$$
2. $$i^{255}$$

1. $$1$$

## Practice Test

##### Exercise $$\PageIndex{33}$$

In the following exercises, simplify using absolute values as necessary.

1. $$\sqrt[3]{125 x^{9}}$$
2. $$\sqrt{169 x^{8} y^{6}}$$
3. $$\sqrt[3]{72 x^{8} y^{4}}$$
4. $$\sqrt{\frac{45 x^{3} y^{4}}{180 x^{5} y^{2}}}$$

1. $$5x^{3}$$

3. $$2 x^{2} y \sqrt[3]{9 x^{2} y}$$

##### Exercise $$\PageIndex{34}$$

In the following exercises, simplify. Assume all variables are positive.

1. $$216^{-\frac{1}{4}}$$
2. $$-49^{\frac{3}{2}}$$
1. $$\sqrt{-45}$$
2. $$\frac{x^{-\frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{-\frac{3}{4}}}$$
3. $$\left(\frac{8 x^{\frac{2}{3}} y^{-\frac{5}{2}}}{x^{-\frac{7}{3}} y^{\frac{1}{2}}}\right)^{\frac{1}{3}}$$
4. $$\sqrt{48 x^{5}}-\sqrt{75 x^{5}}$$
5. $$\sqrt{27 x^{2}}-4 x \sqrt{12}+\sqrt{108 x^{2}}$$
6. $$2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}$$
7. $$\sqrt[3]{4}(\sqrt[3]{16}-\sqrt[3]{6})$$
8. $$(4-3 \sqrt{3})(5+2 \sqrt{3})$$
9. $$\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$$
10. $$\frac{\sqrt{245 x y^{-4}}}{\sqrt{45 x^{4} y^{3}}}$$
11. $$\frac{1}{\sqrt[3]{5}}$$
12. $$\frac{3}{2+\sqrt{3}}$$
13. $$\sqrt{-4} \cdot \sqrt{-9}$$
14. $$-4 i(-2-3 i)$$
15. $$\frac{4+i}{3-2 i}$$
16. $$i^{172}$$

1.

1. $$\frac{1}{4}$$
2. $$-343$$

3. $$x^{\frac{7}{4}}$$

5. $$-x^{2} \sqrt{3 x}$$

7. $$36 x^{4} \sqrt{2}$$

9. $$2-7 \sqrt{3}$$

11. $$\frac{7 x^{5}}{3 y^{7}}$$

13. $$3(2-\sqrt{3})$$

15. $$-12+8i$$

17. $$-i$$

##### Exercise $$\PageIndex{35}$$

In the following exercises, solve.

1. $$\sqrt{2 x+5}+8=6$$
2. $$\sqrt{x+5}+1=x$$
3. $$\sqrt[3]{2 x^{2}-6 x-23}=\sqrt[3]{x^{2}-3 x+5}$$

2. $$x=4$$

##### Exercise $$\PageIndex{36}$$

In the following exercise,

1. find the domain of the function
2. graph the function
3. use the graph to determine the range
1. $$g(x)=\sqrt{x+2}$$
1. domain: $$[-2, \infty)$$
3. range: $$[0, \infty)$$