8.2E: Exercises

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Example $\PageIndex{52}$

$\sqrt{27}$

Answer

$3\sqrt{3}$

Example $\PageIndex{53}$

$\sqrt{80}$

Example $\PageIndex{54}$

$\sqrt{125}$

Answer

$5\sqrt{5}$

Example $\PageIndex{55}$

$\sqrt{96}$

Example $\PageIndex{56}$

$\sqrt{200}$

Answer

$10\sqrt{2}$

Example $\PageIndex{57}$

$\sqrt{147}$

Example $\PageIndex{58}$

$\sqrt{450}$

Answer

$15\sqrt{2}$

Example $\PageIndex{59}$

$\sqrt{252}$

Example $\PageIndex{60}$

$\sqrt{800}$

Answer

$20\sqrt{2}$

Example $\PageIndex{61}$

$\sqrt{288}$

Example $\PageIndex{62}$

$\sqrt{675}$

Answer

$15\sqrt{3}$

Example $\PageIndex{63}$

$\sqrt{1250}$

Example $\PageIndex{64}$

$\sqrt{x^7}$

Answer

$x^3\sqrt{x}$​​​​​​​

Example $\PageIndex{65}$

$\sqrt{y^{11}}$​​​​​​​

Example $\PageIndex{66}$

$\sqrt{p^3}$

Answer

$p\sqrt{p}$​​​​​​​

Example $\PageIndex{67}$

$\sqrt{q^5}$

Example $\PageIndex{68}$

$\sqrt{m^{13}}$

Answer

$m^6\sqrt{m}$​​​​​​​

Example $\PageIndex{69}$

$\sqrt{n^{21}}$​​​​​​​

Example $\PageIndex{70}$

$\sqrt{r^{25}}$

Answer

$r^{12}\sqrt{r}$​​​​​​​

Example $\PageIndex{71}$

$\sqrt{s^{33}}$​​​​​​​

Example $\PageIndex{72}$

$\sqrt{49n^{17}}$

Answer

$7n^8\sqrt{n}$​​​​​​​

Example $\PageIndex{73}$

$\sqrt{25m^9}$​​​​​​​

Example $\PageIndex{74}$

$\sqrt{81r^{15}}$

Answer

$9r^7\sqrt{r}$​​​​​​​

Example $\PageIndex{75}$

$\sqrt{100s^{19}}$​​​​​​​

Example $\PageIndex{76}$

$\sqrt{98m^5}$

Answer

$7m^2\sqrt{2m}$​​​​​​​

Example $\PageIndex{77}$

$\sqrt{32n^{11}}$​​​​​​​

Example $\PageIndex{78}$

$\sqrt{125r^{13}}$

Answer

$5r^6\sqrt{5r}$​​​​​​​

Example $\PageIndex{79}$

$\sqrt{80s^{15}}$

Example $\PageIndex{80}$

$\sqrt{200p^{13}}$

Answer

$10p^6\sqrt{2p}$​​​​​​​

Example $\PageIndex{81}$

$\sqrt{128q^3}$​​​​​​​

Example $\PageIndex{82}$

$\sqrt{242m^{23}}$

Answer

$11m^{11}\sqrt{2m}$​​​​​​​

Example $\PageIndex{83}$

$\sqrt{175n^{13}}$​​​​​​​

Exercise $\PageIndex{84}$

$\sqrt{147m^7n^{11}}$

Answer

$7m^3n^5\sqrt{3mn}$​​​​​​​

Example $\PageIndex{85}$

$\sqrt{48m^7n^5}$​​​​​​​

Example $\PageIndex{86}$

$\sqrt{75r^{13}s^{9}}$

Answer

$5r^{6}s^{4}\sqrt{3rs}$​​​​​​​

Example $\PageIndex{87}$

$\sqrt{96r^3s^3}$​​​​​​​

Example $\PageIndex{88}$

$\sqrt{300p^9q^{11}}$

Answer

$10p^4q^5\sqrt{3pq}$​​​​​​​

Example $\PageIndex{89}$

$\sqrt{192q^3r^7}$​​​​​​​

Example $\PageIndex{90}$

$\sqrt{242m^{13}n^{21}}$

Answer

$11m^6n^{10}\sqrt{2mn}$​​​​​​​

Example $\PageIndex{91}$

$\sqrt{150m^9n^3}$​​​​​​​

Example $\PageIndex{92}$

$5+\sqrt{12}$

Answer

$5+2\sqrt{3}$​​​​​​​

Example $\PageIndex{93}$

$8+\sqrt{96}$​​​​​​​

Example $\PageIndex{94}$

$1+\sqrt{45}$

Answer

$1+3\sqrt{5}$​​​​​​​

Example $\PageIndex{95}$

$3+\sqrt{125}$​​​​​​​

Example $\PageIndex{96}$

$\frac{10−\sqrt{24}}{2}$

Answer

$5−\sqrt{6}$​​​​​​​

Example $\PageIndex{97}$

$\frac{8−\sqrt{80}}{4}$​​​​​​​

Example $\PageIndex{98}$

$\frac{3+\sqrt{90}}{3}$

Answer

$1+\sqrt{10}$​​​​​​​

Example $\PageIndex{99}$

$\frac{15+\sqrt{75}}{5}$

​​​​​​​Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Example $\PageIndex{100}$

$\sqrt{\frac{49}{64}}$

Answer

$\frac{7}{8}$​​​​​​​

Example $\PageIndex{101}$

$\sqrt{\frac{100}{36}}$

Example $\PageIndex{102}$

$\sqrt{\frac{121}{16}}$

Answer

$\frac{11}{4}$

Example $\PageIndex{103}$

$\sqrt{\frac{144}{169}}$

Example $\PageIndex{104}$

$\sqrt{\frac{72}{98}}$

Answer

$\frac{6}{7}$

Example $\PageIndex{105}$

$\sqrt{\frac{75}{12}}$​​​​​​​

Example $\PageIndex{106}$

$\sqrt{\frac{45}{125}}$

Answer

$\frac{3}{5}$

Example $\PageIndex{107}$

$\sqrt{\frac{300}{243}}$

Example $\PageIndex{108}$

$\sqrt{\frac{x^{10}}{x^6}}$

Answer

$x^2$

Example $\PageIndex{109}$

$\sqrt{\frac{p^{20}}{p^{10}}}$​​​​​​​

Example $\PageIndex{110}$

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

Answer

$\frac{1}{y^2}$

Example $\PageIndex{111}$

$\sqrt{\frac{q^8}{q^{14}}}$​​​​​​​

Example $\PageIndex{112}$

$\sqrt{\frac{200x^7}{2x^3}}$

Answer

$10x^2$

Example $\PageIndex{113}$

$\sqrt{\frac{98y^{11}}{2y^5}}$

Example $\PageIndex{114}$

$\sqrt{\frac{96p^9}{6p}}$

Answer

$4p^4$

Example $\PageIndex{115}$

$\sqrt{\frac{108q^{10}}{3q^2}}$​​​​​​​

Example $\PageIndex{116}$

$\sqrt{\frac{36}{35}}$

Answer

$\frac{6}{\sqrt{35}}$

Example $\PageIndex{117}$

$\sqrt{\frac{144}{65}}$

Example $\PageIndex{118}$

$\sqrt{\frac{20}{81}}$

Answer

$\frac{2\sqrt{5}}{9}$​​​​​​​

Example $\PageIndex{119}$

$\sqrt{\frac{211}{96}}$

Example $\PageIndex{120}$

$\sqrt{\frac{96x^7}{121}}$

Answer

$\frac{4x^3\sqrt{6x}}{11}$

Example $\PageIndex{121}$

$\sqrt{\frac{108y^4}{49}}$

Example $\PageIndex{122}$

$\sqrt{\frac{300m^5}{64}}$

Answer

$\frac{5m^2\sqrt{3m}}{4}$

Example $\PageIndex{123}$

$\sqrt{\frac{125n^7}{169}}$​​​​​​​

Example $\PageIndex{124}$

$\sqrt{\frac{98r^5}{100}}$

Answer

$\frac{7r^2\sqrt{2r}}{10}$

Example $\PageIndex{125}$

$\sqrt{\frac{180s^{10}}{144}}$

Example $\PageIndex{126}$

$\sqrt{\frac{28q^6}{225}}$

Answer

$\frac{2q^3\sqrt{7}}{15}$

Example $\PageIndex{127}$

$\sqrt{\frac{150r^3}{256}}$

Example $\PageIndex{128}$

$\sqrt{\frac{75r^9}{s^8}}$

Answer

$\frac{5r^4\sqrt{3r}}{s^4}$

Example $\PageIndex{129}$

$\sqrt{\frac{72x^5}{y^6}}$

Example $\PageIndex{130}$

$\sqrt{\frac{28p^7}{q^2}}$

Answer

$\frac{4p^3\sqrt{7p}}{q}$

Example $\PageIndex{131}$

$\sqrt{\frac{45r^3}{s^{10}}}$

Example $\PageIndex{132}$

$\sqrt{\frac{100x^5}{36x^3}}$

Answer

$\frac{5x}{3}$

Example $\PageIndex{133}$

$\sqrt{\frac{49r^{12}}{16r^6}}$

Example $\PageIndex{134}$

$\sqrt{\frac{121p^5}{81p^2}}$

Answer

$\frac{11p\sqrt{p}}{9}$

Example $\PageIndex{135}$

$\sqrt{\frac{25r^8}{64r}}$​​​​​​​

Example $\PageIndex{136}$

$\sqrt{\frac{32x^{5}y^{3}}{18x^{3}y}}$.

Answer

$\frac{4xy}{3}$​​​​​​​

Example $\PageIndex{137}$

$\sqrt{\frac{75r^{6}s^{8}}{48rs^{4}}}$

Example $\PageIndex{138}$

$\sqrt{\frac{27p^{2}q^{10}}{8p^5q^3}}$

Answer

$\frac{1}{2pq\sqrt{p}}$​​​​​​​

Example $\PageIndex{139}$

$\sqrt{\frac{50r^5s^2}{128r^2s^5}}$

Everyday Math

Example $\PageIndex{140}$

1. Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify $\sqrt{288}$ to determine the length and the width of his garden. Round to the nearest tenth of a foot.
2. Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify $\sqrt{288}−5$ to determine the length and width of the new garden. Round to the nearest tenth of a foot.

Answer

1. 17.0 feet
2. 15.0 feet

Example $\PageIndex{141}$

1. Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify $\sqrt{\frac{64}{16}}$ to determine the number of seconds it takes for the sunglasses to reach the ground.
2. Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify $\sqrt{\frac{144}{16}}$ to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

Example $\PageIndex{142}$

Explain why $\sqrt{x^4}=x^2$. Then explain why $\sqrt{x^{16}}=x^8$.

Answer

Answers will vary.

Example $\PageIndex{143}$

Explain why $7+\sqrt{9}$ is not equal to $\sqrt{7+9}$.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?