17.10: Review Exercises

Chapter Review Exercises

Simplify Expressions with Roots

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

In the following exercises, simplify.

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

Answer

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

Answer

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

Answer

1. Solve for yourself

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

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

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

Answer

1.

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

3.

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

5.

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

Simplify Radical Expressions

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

Answer

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

Answer

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

Answer

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

Answer

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

Answer

1. Solve for yourself

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

In the following exercises, simplify.

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

Answer

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

Answer

1. Solve for yourself

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

In the following exercises, simplify.

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

Answer

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

Answer

1.

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

Add, Subtract and Multiply Radical Expressions

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

In the following exercises, simplify.

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

Answer

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

Answer

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

Answer

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

Divide Radical Expressions

Exercise $\PageIndex{17}$ Divide Square Roots

In the following exercises, simplify.

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

Answer

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

Answer

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. $\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$

Answer

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

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

Solve Radical Equations

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. $2 \sqrt{8 r+1}-8=2$

Answer

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. $\sqrt{x+1}-\sqrt{x-2}=1$

Answer

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.

Answer

2. $64.8$ feet

Use Radicals in Functions

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. $g(-3)$

Answer

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. $F(x)=\sqrt[4]{10-7 x}$

Answer

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. $f(x)=\sqrt[3]{x}+3$

Answer

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

Answer

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. $(-7-\sqrt{-50})-(-32-\sqrt{-18})$

Answer

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. $(5-\sqrt{-12})(-3+\sqrt{-75})$

Answer

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

Answer

1. $-5-12 i$

Exercise $\PageIndex{30}$ Multiply Complex Numbers

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

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

Answer

Solve for yourself

Exercise $\PageIndex{31}$ divide Complex Numbers

In the following exercises, divide.

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

Answer

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

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

In the following exercises, simplify.

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

Answer

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. $\sqrt{\frac{45 x^{3} y^{4}}{180 x^{5} y^{2}}}$

Answer

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

Answer

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

Answer

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. $g(x)=\sqrt{x+2}$

Answer

1.

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