13.8.9: Chapter 9 Review Exercises

Chapter 9 Review Exercises

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{1}$

$\sqrt{64}$

Exercise $\PageIndex{2}$

$\sqrt{144}$

Answer

12

Exercise $\PageIndex{3}$

$−\sqrt{25}$

Exercise $\PageIndex{4}$

$−\sqrt{81}$

Answer

−9

Exercise $\PageIndex{5}$

$\sqrt{−9}$

Exercise $\PageIndex{6}$

$\sqrt{−36}$

Answer

not a real number

Exercise $\PageIndex{7}$

$\sqrt{64}+\sqrt{225}$

Exercise $\PageIndex{8}$

$\sqrt{64+225}$

Answer

17

Estimate Square Roots

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

Exercise $\PageIndex{9}$

$\sqrt{28}$

Exercise $\PageIndex{10}$

$\sqrt{155}$

Answer

$12<\sqrt{155}<13$

Approximate Square Roots

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

Exercise $\PageIndex{11}$

$\sqrt{15}$

Example $\PageIndex{12}$

$\sqrt{57}$

Answer

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{13}$

$\sqrt{q^2}$

Exercise $\PageIndex{14}$

$\sqrt{64b^2}$

Answer

8b

Exercis\e $\PageIndex{15}$

$−\sqrt{121a^2}$

Exercise $\PageIndex{16}$

$\sqrt{225m^{2}n^{2}}$

Answer

15mn

Exercise $\PageIndex{17}$

$−\sqrt{100q^2}$

Exercise $\PageIndex{18}$

$\sqrt{49y^2}$

Answer

7y

Exercise $\PageIndex{19}$

$\sqrt{4a^{2}b^{2}}$

Exercise $\PageIndex{20}$

$\sqrt{121c^{2}d^{2}}$

Answer

11cd

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{21}$

$\sqrt{300}$

Exercise $\PageIndex{22}$

$\sqrt{98}$

Answer

$7\sqrt{2}$

Exercise $\PageIndex{23}$

$\sqrt{x^{13}}$

Exercise $\PageIndex{24}$

$\sqrt{y^{19}}$

Answer

$y^{9}\sqrt{y}$

Exercise $\PageIndex{25}$

$\sqrt{16m^4}$

Exercise $\PageIndex{26}$

$\sqrt{36n^{13}}$

Answer

$6n^{6}\sqrt{n}$

Exercise $\PageIndex{27}$

$\sqrt{288m^{21}}$

Exercise $\PageIndex{28}$

$\sqrt{150n^7}$

Answer

$5n^3\sqrt{6n}$

Exercise $\PageIndex{29}$

$\sqrt{48r^{5}s^{4}}$

Exercise $\PageIndex{30}$

$\sqrt{108r^{5}s^{3}}$

Answer

$6r^{2}s\sqrt{3rs}$

Exercise $\PageIndex{31}$

$\frac{10−\sqrt{50}}{5}$

Exercise $\PageIndex{32}$

$\frac{6+\sqrt{72}}{6}$

Answer

$1+\sqrt{2}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{33}$

$\sqrt{\frac{16}{25}}$

Exercise $\PageIndex{34}$

$\sqrt{\frac{81}{36}}$

Answer

$\frac{3}{2}$

Exercise $\PageIndex{35}$

$\sqrt{\frac{x^8}{x^4}}$

Exercise $\PageIndex{36}$

$\sqrt{\frac{y^6}{y^2}}$

Answer

$y^2$

Exercise $\PageIndex{37}$

$\sqrt{\frac{98p^6}{2p^2}}$

Exercise $\PageIndex{38}$

$\sqrt{\frac{72q^8}{2q^4}}$

Answer

$6q^2$

Exercise $\PageIndex{39}$

$\sqrt{\frac{65}{121}}$

Exercise $\PageIndex{40}$

$\sqrt{\frac{26}{169}}$

Answer

$\frac{\sqrt{26}}{13}$

Exercise $\PageIndex{41}$

$\sqrt{\frac{64x^4}{25x^2}}$

Exercise $\PageIndex{42}$

$\sqrt{\frac{36r^{10}}{16r^5}}$

Answer

$\frac{3r^2\sqrt{r}}{2}$

Exercise $\PageIndex{43}$

$\sqrt{\frac{48p^{3}q^{5}}{27pq}}$

Exercise $\PageIndex{44}$

$\sqrt{\frac{12r^{5}s^{7}}{75r^{2}s}}$

Answer

$\frac{2rs^3\sqrt{r}}{5}$

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{45}$

$3\sqrt{2}+\sqrt{2}$

Exercise $\PageIndex{46}$

$5\sqrt{5}+7\sqrt{5}$

Answer

$12\sqrt{5}$

Exercise $\PageIndex{47}$

$4\sqrt{y}+4\sqrt{y}$

Exercise $\PageIndex{48}$

$6\sqrt{m}−2\sqrt{m}$

Answer

$4\sqrt{m}$

Exercise $\PageIndex{49}$

$−3\sqrt{7}+2\sqrt{7}−\sqrt{7}$

Exercise $\PageIndex{50}$

$8\sqrt{13}+2\sqrt{3}+3\sqrt{13}$

Answer

$11\sqrt{13}+2\sqrt{3}$

Exercise $\PageIndex{51}$

$3\sqrt{5xy}−\sqrt{5xy}+3\sqrt{5xy}$

Exercise $\PageIndex{52}$

$2\sqrt{3rs}+\sqrt{3rs}−5\sqrt{rs}$

Answer

$3\sqrt{3rs}−5\sqrt{rs}$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Exercise $\PageIndex{53}$

$\sqrt{32}+3\sqrt{2}$

Exercise $\PageIndex{54}$

$\sqrt{8}+\sqrt{32}$

Answer

$5\sqrt{2}$

Exercise $\PageIndex{55}$

$\sqrt{72}+\sqrt{50}$

Exercise $\PageIndex{56}$

$\sqrt{48}+\sqrt{75}$

Answer

$9\sqrt{3}$

Exercise $\PageIndex{57}$

$3\sqrt{32}+\sqrt{98}$

Exercise $\PageIndex{58}$

$\frac{1}{3}\sqrt{27}−\frac{1}{8}\sqrt{192}$

Answer

0

Exercise $\PageIndex{59}$

$\sqrt{50y^5}−\sqrt{72y^5}$

Exercise $\PageIndex{60}$

Add exercises text here.

Answer

$17n^2\sqrt{2}$

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{61}$

$\sqrt{2}·\sqrt{20}$

Exercise $\PageIndex{62}$

$2\sqrt{2}·6\sqrt{14}$

Answer

$24\sqrt{7}$

Exercise $\PageIndex{63}$

$\sqrt{2m^2}·\sqrt{20m^4}$

Exercise $\PageIndex{64}$

$(\sqrt{62y})(\sqrt{350y^3})$

Answer

$180y^2$

Exercise $\PageIndex{65}$

$(6\sqrt{3v^4})(5\sqrt{30v})$

Exercise $\PageIndex{66}$

$(\sqrt{8})^2$

Answer

8

Exercise $\PageIndex{67}$

$(−\sqrt{10})^2$

Exercise $\PageIndex{68}$

$(2\sqrt{5})(5\sqrt{5})$

Answer

50

Exercise $\PageIndex{69}$

$(−3\sqrt{3})(5\sqrt{18})$

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{70}$

$10(2−\sqrt{7})$

Answer

$20−10\sqrt{7}$

Exercise $\PageIndex{71}$

$\sqrt{3}(4+\sqrt{12})$

Exercise $\PageIndex{72}$

$(5+\sqrt{2})(3−\sqrt{2})$

Answer

$13−2\sqrt{2}$

Exercise $\PageIndex{73}$

$(5−3\sqrt{7})(1−2\sqrt{7})$

Exercise $\PageIndex{74}$

$(1−3\sqrt{x})(5+2\sqrt{x})$

Answer

$5−13\sqrt{x}−6x$

Exercise $\PageIndex{75}$

$(3+4\sqrt{y})(10−\sqrt{y})$

Exercise $\PageIndex{76}$

$(1+6\sqrt{p})^2$

Answer

$1+12\sqrt{p}+36p$

Exercise $\PageIndex{77}$

$(2−6\sqrt{5})^2$

Exercise $\PageIndex{78}$

$(3+2\sqrt{7})(3−2\sqrt{7})$

Answer

−19

Exercise $\PageIndex{79}$

$(6−\sqrt{11})(6+\sqrt{11})$

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

Exercise $\PageIndex{80}$

$\frac{\sqrt{75}}{10}$

Answer

$\frac{\sqrt{3}}{2}$

Exercise $\PageIndex{81}$

$\frac{2−\sqrt{12}}{6}$

Exercise $\PageIndex{82}$

$\frac{\sqrt{48}}{\sqrt{27}}$

Answer

$\frac{4}{3}$

Exercise $\PageIndex{83}$

$\frac{\sqrt{75x^7}}{\sqrt{3x^3}}$

Exercise $\PageIndex{84}$

$\frac{\sqrt{20y^5}}{\sqrt{2y}}$

Answer

$y^2\sqrt{10}$

Exercise $\PageIndex{85}$

$\frac{\sqrt{98p^{6}q^{4}}}{\sqrt{2p^{4}q^{8}}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Exercise $\PageIndex{86}$

$\frac{10}{\sqrt{15}}$

Answer

$\frac{2\sqrt{15}}{3}$

Exercise $\PageIndex{87}$

$\frac{6}{\sqrt{6}}$

Exercise $\PageIndex{88}$

$\frac{5}{3\sqrt{5}}$

Answer

$\frac{\sqrt{5}}{3}$

Exercise $\PageIndex{89}$

$\frac{10}{2\sqrt{6}}$

Exercise $\PageIndex{90}$

$\sqrt{\frac{3}{28}}$

Answer

$\frac{\sqrt{21}}{14}$

Exercise $\PageIndex{91}$

$\sqrt{\frac{9}{75}}$

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

Exercise $\PageIndex{92}$

$\frac{4}{4+\sqrt{27}}$

Answer

$\frac{16−12\sqrt{3}}{−11}$

Exercise $\PageIndex{93}$

$\frac{5}{2−\sqrt{10}}$

Exercise $\PageIndex{94}$

$\frac{4}{2−\sqrt{5}}$

Answer

$−8−4\sqrt{5}$

Exercise $\PageIndex{95}$

$\frac{5}{4−\sqrt{8}}$

Exercise $\PageIndex{96}$

$\frac{\sqrt{2}}{\sqrt{p}+\sqrt{3}}$

Answer

$\frac{\sqrt{2p}−\sqrt{6}}{p−3}$

Exercise $\PageIndex{97}$

$\frac{\sqrt{x}−\sqrt{2}}{\sqrt{x}+\sqrt{2}}$

Solve Equations with Square Roots

Solve Radical Equations

In the following exercises, solve the equation.

Exercise $\PageIndex{98}$

$\sqrt{7z+1}=6$

Answer

5

Exercise $\PageIndex{99}$

$\sqrt{4u−2}−4=0$

Exercise $\PageIndex{100}$

$\sqrt{6m+4}−5=0$

Answer

$\frac{7}{2}$

Exercise $\PageIndex{101}$

$\sqrt{2u−3}+2=0$

Exercise $\PageIndex{102}$

$\sqrt{u−4}+4=u$

Answer

no solution

Exercise $\PageIndex{103}$

$\sqrt{v−9}+9=0$​​​​​​​

Exercise $\PageIndex{104}$

$\sqrt{r−4}−r=−10$

Answer

13

Exercise $\PageIndex{105}$

$\sqrt{s−9}−s=−9$

Exercise $\PageIndex{106}$

$2\sqrt{2x−7}−4=8$

Answer

$\frac{43}{2}$

Exercise $\PageIndex{107}$

$\sqrt{2−x}=\sqrt{2x−7}$

Exercise $\PageIndex{108}$

$\sqrt{a}+3=\sqrt{a+9}$

Answer

0

Exercise $\PageIndex{109}$

$\sqrt{r}+3=\sqrt{r+4}$​​​​​​​

Exercise $\PageIndex{110}$

$\sqrt{u}+2=\sqrt{u+5}$

Answer

$\frac{11}{6}$

Exercise $\PageIndex{111}$

$\sqrt{n+11}−1=\sqrt{n+4}$

Exercise $\PageIndex{112}$

$\sqrt{y+5}+1=\sqrt{2y+3}$

Answer

11

​​​​​​​Use Square Roots in Applications

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

Exercise $\PageIndex{113}$

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula $s=\sqrt{A}$ to find the length of each side of his lawn.​​​​​​​

Exercise $\PageIndex{114}$

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

Answer

7.5 seconds

Exercise $\PageIndex{115}$

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the car before the brakes were applied.

Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $\PageIndex{116}$

1. $\sqrt[6]{64}$
2. $\sqrt[3]{64}$

Answer

1. 2
2. 4

Exercise $\PageIndex{117}$

1. $\sqrt[3]{−27}$
2. $\sqrt[4]{−64}$

Exercise $\PageIndex{118}$

1. $\sqrt[9]{d^9}$
2. $\sqrt[8]{v^8}$

Answer

1. d
2. |v|

Exercise $\PageIndex{119}$

1. $\sqrt[5]{a^{10}}$
2. $\sqrt[3]{b^{27}}$

Exercise $\PageIndex{120}$

1. $\sqrt[4]{16x^8}$
2. $\sqrt[6]{64y^{12}}$

Answer

1. $2x^2$
2. $2y^2$

Exercise $\PageIndex{121}$

1. $\sqrt[7]{128r^{14}}$
2. $\sqrt[4]{81s^{24}}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $\PageIndex{122}$

1. $\sqrt[9]{d^9}$

Answer

1. d

​​​​​​​​​​​​​​

Exercise $\PageIndex{123}$

1. $\sqrt[3]{54}$
2. $\sqrt[4]{128}$

Exercise $\PageIndex{124}$

1. $\sqrt[5]{64c^8}$
2. $\sqrt[4]{48d^7}$

Answer

1. $2c\sqrt[5]{2c^3}$
2. $2d\sqrt[4]{3d^3}$

Exercise $\PageIndex{125}$

1. $\sqrt[3]{343q^7}$
2. $\sqrt[6]{192r^9}$

Exercise $\PageIndex{126}$

1. $\sqrt[3]{−500}$
2. $\sqrt[4]{−16}$

Answer

1. $−5\sqrt[3]{4}$
2. not a real number

​​​​​​​Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $\PageIndex{127}$

$\sqrt[5]{\frac{r^{10}}{r^5}}$

Exercise $\PageIndex{128}$

$\sqrt[3]{\frac{w^{12}}{w^2}}$

Answer

$w^3\sqrt[3]{w}$

Exercise $\PageIndex{129}$

$\sqrt[4]{\frac{64y^8}{4y^5}}$

Exercise $\PageIndex{130}$

$\sqrt[3]{\frac{54z^9}{2z^3}}$

Answer

$3z^2$

Exercise $\PageIndex{131}$

$\sqrt[6]{\frac{64a^7}{b^2}}$

​​​​​​​Add and Subtract Higher Roots

In the following exercises, simplify.

Exercise $\PageIndex{132}$

$4\sqrt[5]{20}−2\sqrt[5]{20}$

Answer

$2\sqrt[5]{20}$

Exercise $\PageIndex{133}$

$4\sqrt[3]{18}+3\sqrt[3]{18}$

Exercise $\PageIndex{134}$

$\sqrt[4]{1250}−\sqrt[4]{162}$

Answer

$2\sqrt[4]{2}$

Exercise $\PageIndex{135}$

$\sqrt[3]{640c^5}−\sqrt[3]{−80c^3}$

Exercise $\PageIndex{136}$

$\sqrt[5]{96t^8}+\sqrt[5]{486t^4}$

Answer

$2t^\sqrt[5]{3t^3}+3\sqrt[5]{2t^4}$

Rational Exponents

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

In the following exercises, write as a radical expression.

Exercise $\PageIndex{137}$

$r^{\frac{1}{8}}$

Exercise $\PageIndex{138}$

$s^{\frac{1}{10}}$

Answer

​​​​​​​In the following exercises, write with a rational exponent.

Exercise $\PageIndex{139}$

$\sqrt[5]{u}$

Exercise $\PageIndex{140}$

$\sqrt[6]{v}$

Answer

$v^{\frac{1}{6}}$

Exercise $\PageIndex{141}$

$\sqrt[3]{9m}$

Exercise $\PageIndex{142}$

$\sqrt[6]{10z}$

Answer

$(10z)^{\frac{1}{6}}$

​​​​​​​In the following exercises, simplify.

Exercise $\PageIndex{143}$

$16^{\frac{1}{4}}$

Exercise $\PageIndex{144}$

$32^{\frac{1}{5}}$

Answer

2

Exercise $\PageIndex{145}$

$(−125)^{\frac{1}{3}}$

Exercise $\PageIndex{146}$

$(125)^{−\frac{1}{3}}$

Answer

$\frac{1}{5}$

Exercise $\PageIndex{147}$

$(−9)^{\frac{1}{2}}$

Exercise $\PageIndex{148}$

$(36)^{−\frac{1}{2}}$

Answer

$\frac{1}{6}$

​​​​​​​Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

Exercise $\PageIndex{149}$

$\sqrt[3]{q^5}$

Exercise $\PageIndex{150}$

$\sqrt[5]{n^8}$

Answer

$n^{\frac{8}{5}}$

In the following exercises, simplify.

Exercise $\PageIndex{151}$

$27^{−\frac{2}{3}}$

Exercise $\PageIndex{152}$

$64^{\frac{5}{2}}$

Answer

32,768

Exercise $\PageIndex{153}$

$36^{\frac{3}{2}}$

Exercise $\PageIndex{154}$

$81^{−\frac{5}{2}}$

Answer

$\frac{1}{59,049}$

​​​​​​​Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Exercise $\PageIndex{155}$

$3^{\frac{4}{5}}·3^{\frac{6}{5}}$

Exercise $\PageIndex{156}$

$(x^6)^{\frac{4}{3}}$

Answer

$x^8$

Exercise $\PageIndex{157}$

$\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}$

Exercise $\PageIndex{158}$

$(16s^{\frac{9}{4}})^{\frac{1}{4}}$

Answer

$2s^{\frac{9}{16}}$

Exercise $\PageIndex{159}$

$(m^{8}n^{12})^{\frac{1}{4}}$

Exercise $\PageIndex{160}$

$\frac{z^{\frac{2}{3}}·z^{−\frac{1}{3}}}{z^{−\frac{5}{3}}}$

Answer

$z^2$​​​​​​​

Practice Test

In the following exercises, simplify.

Exercise $\PageIndex{161}$

$\sqrt{81+144}$

Exercise $\PageIndex{162}$

$\sqrt{169m^{4}n^{2}}$

Answer

$13m^{2}|n|$

Exercise $\PageIndex{163}$

$\sqrt{36n^{13}}$

Exercise $\PageIndex{164}$

$3\sqrt{13}+5\sqrt{2}+\sqrt{13}$

Answer

$4\sqrt{13}+5\sqrt{2}$

Exercise $\PageIndex{165}$

$5\sqrt{20}+2\sqrt{125}$

Exercise $\PageIndex{166}$

$(3\sqrt{6y})(\sqrt{250y^3})$

Answer

$180y^2\sqrt{3}$

Exercise $\PageIndex{167}$

$(2−5\sqrt{x})(3+\sqrt{x})$

Exercise $\PageIndex{168}$

$(1−2\sqrt{q})^2$

Answer

$1−4\sqrt{q}+4q$

Exercise $\PageIndex{169}$

1. $\sqrt{a^{12}}$
2. $\sqrt[3]{b^{21}}$

Exercise $\PageIndex{170}$

1. $\sqrt[4]{81x^{12}}$
2. $\sqrt[6]{64y^{18}}$

Answer

1. $3x^3$
2. $2y^3$

Exercise $\PageIndex{171}$

$\sqrt[6]{\frac{64r^{12}}{25r^6}}$​​​​​​​

Exercise $\PageIndex{172}$

$\sqrt{\frac{14y^3}{7y}}$

Answer

$y\sqrt{2}$

Exercise $\PageIndex{173}$

$\frac{\sqrt{256x^7}}{\sqrt{54x^2}}$

Exercise $\PageIndex{174}$

$\sqrt[4]{512}−2\sqrt[4]{32}$

Answer

0

Exercise $\PageIndex{175}$

1. $256^{\frac{1}{4}}$
2. $243^{\frac{1}{5}}$

Exercise $\PageIndex{176}$

$49^{\frac{3}{2}}$

Answer

343

Exercise $\PageIndex{177}$

$25^{−\frac{5}{2}}$

Exercise $\PageIndex{178}$

$\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}$

Answer

$\frac{1}{w}$

Exercise $\PageIndex{179}$

$(27s^{\frac{3}{5}})^{\frac{1}{3}}$

​​​​​​​In the following exercises, rationalize the denominator.

Exercise $\PageIndex{180}$

$\frac{3}{2\sqrt{6}}$

Answer

$\frac{\sqrt{6}}{4}$

Exercise $\PageIndex{181}$

$\frac{\sqrt{3}}{\sqrt{x}+\sqrt{5}}$

In the following exercises, solve.

Exercise $\PageIndex{182}$

$3\sqrt{2x−3}−20=7$

Answer

42​​​​​​​

Exercise $\PageIndex{183}$

$\sqrt{3u−2}=\sqrt{5u+1}$

​​​​​​​In the following exercise, solve.

Exercise $\PageIndex{184}$

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

Answer

6.1 seconds