13.8.2E: Exercises
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Practice Makes Perfect
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Example \(\PageIndex{52}\)
\(\sqrt{27}\)
- Answer
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\(3\sqrt{3}\)
Example \(\PageIndex{53}\)
\(\sqrt{80}\)
Example \(\PageIndex{54}\)
\(\sqrt{125}\)
- Answer
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\(5\sqrt{5}\)
Example \(\PageIndex{55}\)
\(\sqrt{96}\)
Example \(\PageIndex{56}\)
\(\sqrt{200}\)
- Answer
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\(10\sqrt{2}\)
Example \(\PageIndex{57}\)
\(\sqrt{147}\)
Example \(\PageIndex{58}\)
\(\sqrt{450}\)
- Answer
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\(15\sqrt{2}\)
Example \(\PageIndex{59}\)
\(\sqrt{252}\)
Example \(\PageIndex{60}\)
\(\sqrt{800}\)
- Answer
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\(20\sqrt{2}\)
Example \(\PageIndex{61}\)
\(\sqrt{288}\)
Example \(\PageIndex{62}\)
\(\sqrt{675}\)
- Answer
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\(15\sqrt{3}\)
Example \(\PageIndex{63}\)
\(\sqrt{1250}\)
Example \(\PageIndex{64}\)
\(\sqrt{x^7}\)
- Answer
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\(x^3\sqrt{x}\)
Example \(\PageIndex{65}\)
\(\sqrt{y^{11}}\)
Example \(\PageIndex{66}\)
\(\sqrt{p^3}\)
- Answer
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\(p\sqrt{p}\)
Example \(\PageIndex{67}\)
\(\sqrt{q^5}\)
Example \(\PageIndex{68}\)
\(\sqrt{m^{13}}\)
- Answer
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\(m^6\sqrt{m}\)
Example \(\PageIndex{69}\)
\(\sqrt{n^{21}}\)
Example \(\PageIndex{70}\)
\(\sqrt{r^{25}}\)
- Answer
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\(r^{12}\sqrt{r}\)
Example \(\PageIndex{71}\)
\(\sqrt{s^{33}}\)
Example \(\PageIndex{72}\)
\(\sqrt{49n^{17}}\)
- Answer
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\(7n^8\sqrt{n}\)
Example \(\PageIndex{73}\)
\(\sqrt{25m^9}\)
Example \(\PageIndex{74}\)
\(\sqrt{81r^{15}}\)
- Answer
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\(9r^7\sqrt{r}\)
Example \(\PageIndex{75}\)
\(\sqrt{100s^{19}}\)
Example \(\PageIndex{76}\)
\(\sqrt{98m^5}\)
- Answer
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\(7m^2\sqrt{2m}\)
Example \(\PageIndex{77}\)
\(\sqrt{32n^{11}}\)
Example \(\PageIndex{78}\)
\(\sqrt{125r^{13}}\)
- Answer
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\(5r^6\sqrt{5r}\)
Example \(\PageIndex{79}\)
\(\sqrt{80s^{15}}\)
Example \(\PageIndex{80}\)
\(\sqrt{200p^{13}}\)
- Answer
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\(10p^6\sqrt{2p}\)
Example \(\PageIndex{81}\)
\(\sqrt{128q^3}\)
Example \(\PageIndex{82}\)
\(\sqrt{242m^{23}}\)
- Answer
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\(11m^{11}\sqrt{2m}\)
Example \(\PageIndex{83}\)
\(\sqrt{175n^{13}}\)
Exercise \(\PageIndex{84}\)
\(\sqrt{147m^7n^{11}}\)
- Answer
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\(7m^3n^5\sqrt{3mn}\)
Example \(\PageIndex{85}\)
\(\sqrt{48m^7n^5}\)
Example \(\PageIndex{86}\)
\(\sqrt{75r^{13}s^{9}}\)
- Answer
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\(5r^{6}s^{4}\sqrt{3rs}\)
Example \(\PageIndex{87}\)
\(\sqrt{96r^3s^3}\)
Example \(\PageIndex{88}\)
\(\sqrt{300p^9q^{11}}\)
- Answer
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\(10p^4q^5\sqrt{3pq}\)
Example \(\PageIndex{89}\)
\(\sqrt{192q^3r^7}\)
Example \(\PageIndex{90}\)
\(\sqrt{242m^{13}n^{21}}\)
- Answer
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\(11m^6n^{10}\sqrt{2mn}\)
Example \(\PageIndex{91}\)
\(\sqrt{150m^9n^3}\)
Example \(\PageIndex{92}\)
\(5+\sqrt{12}\)
- Answer
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\(5+2\sqrt{3}\)
Example \(\PageIndex{93}\)
\(8+\sqrt{96}\)
Example \(\PageIndex{94}\)
\(1+\sqrt{45}\)
- Answer
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\(1+3\sqrt{5}\)
Example \(\PageIndex{95}\)
\(3+\sqrt{125}\)
Example \(\PageIndex{96}\)
\(\frac{10−\sqrt{24}}{2}\)
- Answer
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\(5−\sqrt{6}\)
Example \(\PageIndex{97}\)
\(\frac{8−\sqrt{80}}{4}\)
Example \(\PageIndex{98}\)
\(\frac{3+\sqrt{90}}{3}\)
- Answer
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\(1+\sqrt{10}\)
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
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\(\frac{7}{8}\)
Example \(\PageIndex{101}\)
\(\sqrt{\frac{100}{36}}\)
Example \(\PageIndex{102}\)
\(\sqrt{\frac{121}{16}}\)
- Answer
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\(\frac{11}{4}\)
Example \(\PageIndex{103}\)
\(\sqrt{\frac{144}{169}}\)
Example \(\PageIndex{104}\)
\(\sqrt{\frac{72}{98}}\)
- Answer
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\(\frac{6}{7}\)
Example \(\PageIndex{105}\)
\(\sqrt{\frac{75}{12}}\)
Example \(\PageIndex{106}\)
\(\sqrt{\frac{45}{125}}\)
- Answer
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\(\frac{3}{5}\)
Example \(\PageIndex{107}\)
\(\sqrt{\frac{300}{243}}\)
Example \(\PageIndex{108}\)
\(\sqrt{\frac{x^{10}}{x^6}}\)
- Answer
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\(x^2\)
Example \(\PageIndex{109}\)
\(\sqrt{\frac{p^{20}}{p^{10}}}\)
Example \(\PageIndex{110}\)
\(\sqrt{\frac{y^4}{y^8}}\)
- Answer
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\(\frac{1}{y^2}\)
Example \(\PageIndex{111}\)
\(\sqrt{\frac{q^8}{q^{14}}}\)
Example \(\PageIndex{112}\)
\(\sqrt{\frac{200x^7}{2x^3}}\)
- Answer
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\(10x^2\)
Example \(\PageIndex{113}\)
\(\sqrt{\frac{98y^{11}}{2y^5}}\)
Example \(\PageIndex{114}\)
\(\sqrt{\frac{96p^9}{6p}}\)
- Answer
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\(4p^4\)
Example \(\PageIndex{115}\)
\(\sqrt{\frac{108q^{10}}{3q^2}}\)
Example \(\PageIndex{116}\)
\(\sqrt{\frac{36}{35}}\)
- Answer
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\(\frac{6}{\sqrt{35}}\)
Example \(\PageIndex{117}\)
\(\sqrt{\frac{144}{65}}\)
Example \(\PageIndex{118}\)
\(\sqrt{\frac{20}{81}}\)
- Answer
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\(\frac{2\sqrt{5}}{9}\)
Example \(\PageIndex{119}\)
\(\sqrt{\frac{211}{96}}\)
Example \(\PageIndex{120}\)
\(\sqrt{\frac{96x^7}{121}}\)
- Answer
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\(\frac{4x^3\sqrt{6x}}{11}\)
Example \(\PageIndex{121}\)
\(\sqrt{\frac{108y^4}{49}}\)
Example \(\PageIndex{122}\)
\(\sqrt{\frac{300m^5}{64}}\)
- Answer
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\(\frac{5m^2\sqrt{3m}}{4}\)
Example \(\PageIndex{123}\)
\(\sqrt{\frac{125n^7}{169}}\)
Example \(\PageIndex{124}\)
\(\sqrt{\frac{98r^5}{100}}\)
- Answer
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\(\frac{7r^2\sqrt{2r}}{10}\)
Example \(\PageIndex{125}\)
\(\sqrt{\frac{180s^{10}}{144}}\)
Example \(\PageIndex{126}\)
\(\sqrt{\frac{28q^6}{225}}\)
- Answer
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\(\frac{2q^3\sqrt{7}}{15}\)
Example \(\PageIndex{127}\)
\(\sqrt{\frac{150r^3}{256}}\)
Example \(\PageIndex{128}\)
\(\sqrt{\frac{75r^9}{s^8}}\)
- Answer
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\(\frac{5r^4\sqrt{3r}}{s^4}\)
Example \(\PageIndex{129}\)
\(\sqrt{\frac{72x^5}{y^6}}\)
Example \(\PageIndex{130}\)
\(\sqrt{\frac{28p^7}{q^2}}\)
- Answer
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\(\frac{4p^3\sqrt{7p}}{q}\)
Example \(\PageIndex{131}\)
\(\sqrt{\frac{45r^3}{s^{10}}}\)
Example \(\PageIndex{132}\)
\(\sqrt{\frac{100x^5}{36x^3}}\)
- Answer
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\(\frac{5x}{3}\)
Example \(\PageIndex{133}\)
\(\sqrt{\frac{49r^{12}}{16r^6}}\)
Example \(\PageIndex{134}\)
\(\sqrt{\frac{121p^5}{81p^2}}\)
- Answer
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\(\frac{11p\sqrt{p}}{9}\)
Example \(\PageIndex{135}\)
\(\sqrt{\frac{25r^8}{64r}}\)
Example \(\PageIndex{136}\)
\(\sqrt{\frac{32x^{5}y^{3}}{18x^{3}y}}\).
- Answer
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\(\frac{4xy}{3}\)
Example \(\PageIndex{137}\)
\(\sqrt{\frac{75r^{6}s^{8}}{48rs^{4}}}\)
Example \(\PageIndex{138}\)
\(\sqrt{\frac{27p^{2}q^{10}}{8p^5q^3}}\)
- Answer
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\(\frac{1}{2pq\sqrt{p}}\)
Example \(\PageIndex{139}\)
\(\sqrt{\frac{50r^5s^2}{128r^2s^5}}\)
Everyday Math
Example \(\PageIndex{140}\)
- 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.
- 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
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- 17.0 feet
- 15.0 feet
Example \(\PageIndex{141}\)
- 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.
- 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
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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?