8.2E: Exercises
- Page ID
- 79516
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
\(\sqrt{27}\)
- Answer
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\(3\sqrt{3}\)
\(\sqrt{80}\)
\(\sqrt{125}\)
- Answer
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\(5\sqrt{5}\)
\(\sqrt{96}\)
\(\sqrt{200}\)
- Answer
-
\(10\sqrt{2}\)
\(\sqrt{147}\)
\(\sqrt{450}\)
- Answer
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\(15\sqrt{2}\)
\(\sqrt{252}\)
\(\sqrt{800}\)
- Answer
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\(20\sqrt{2}\)
\(\sqrt{288}\)
\(\sqrt{675}\)
- Answer
-
\(15\sqrt{3}\)
\(\sqrt{1250}\)
\(\sqrt{x^7}\)
- Answer
-
\(x^3\sqrt{x}\)
\(\sqrt{y^{11}}\)
\(\sqrt{p^3}\)
- Answer
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\(p\sqrt{p}\)
\(\sqrt{q^5}\)
\(\sqrt{m^{13}}\)
- Answer
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\(m^6\sqrt{m}\)
\(\sqrt{n^{21}}\)
\(\sqrt{r^{25}}\)
- Answer
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\(r^{12}\sqrt{r}\)
\(\sqrt{s^{33}}\)
\(\sqrt{49n^{17}}\)
- Answer
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\(7n^8\sqrt{n}\)
\(\sqrt{25m^9}\)
\(\sqrt{81r^{15}}\)
- Answer
-
\(9r^7\sqrt{r}\)
\(\sqrt{100s^{19}}\)
\(\sqrt{98m^5}\)
- Answer
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\(7m^2\sqrt{2m}\)
\(\sqrt{32n^{11}}\)
\(\sqrt{125r^{13}}\)
- Answer
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\(5r^6\sqrt{5r}\)
\(\sqrt{80s^{15}}\)
\(\sqrt{200p^{13}}\)
- Answer
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\(10p^6\sqrt{2p}\)
\(\sqrt{128q^3}\)
\(\sqrt{242m^{23}}\)
- Answer
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\(11m^{11}\sqrt{2m}\)
\(\sqrt{175n^{13}}\)
\(\sqrt{147m^7n^{11}}\)
- Answer
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\(7m^3n^5\sqrt{3mn}\)
\(\sqrt{48m^7n^5}\)
\(\sqrt{75r^{13}s^{9}}\)
- Answer
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\(5r^{6}s^{4}\sqrt{3rs}\)
\(\sqrt{96r^3s^3}\)
\(\sqrt{300p^9q^{11}}\)
- Answer
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\(10p^4q^5\sqrt{3pq}\)
\(\sqrt{192q^3r^7}\)
\(\sqrt{242m^{13}n^{21}}\)
- Answer
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\(11m^6n^{10}\sqrt{2mn}\)
\(\sqrt{150m^9n^3}\)
\(5+\sqrt{12}\)
- Answer
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\(5+2\sqrt{3}\)
\(8+\sqrt{96}\)
\(1+\sqrt{45}\)
- Answer
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\(1+3\sqrt{5}\)
\(3+\sqrt{125}\)
\(\frac{10−\sqrt{24}}{2}\)
- Answer
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\(5−\sqrt{6}\)
\(\frac{8−\sqrt{80}}{4}\)
\(\frac{3+\sqrt{90}}{3}\)
- Answer
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\(1+\sqrt{10}\)
\(\frac{15+\sqrt{75}}{5}\)
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.
\(\sqrt{\frac{49}{64}}\)
- Answer
-
\(\frac{7}{8}\)
\(\sqrt{\frac{100}{36}}\)
\(\sqrt{\frac{121}{16}}\)
- Answer
-
\(\frac{11}{4}\)
\(\sqrt{\frac{144}{169}}\)
\(\sqrt{\frac{72}{98}}\)
- Answer
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\(\frac{6}{7}\)
\(\sqrt{\frac{75}{12}}\)
\(\sqrt{\frac{45}{125}}\)
- Answer
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\(\frac{3}{5}\)
\(\sqrt{\frac{300}{243}}\)
\(\sqrt{\frac{x^{10}}{x^6}}\)
- Answer
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\(x^2\)
\(\sqrt{\frac{p^{20}}{p^{10}}}\)
\(\sqrt{\frac{y^4}{y^8}}\)
- Answer
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\(\frac{1}{y^2}\)
\(\sqrt{\frac{q^8}{q^{14}}}\)
\(\sqrt{\frac{200x^7}{2x^3}}\)
- Answer
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\(10x^2\)
\(\sqrt{\frac{98y^{11}}{2y^5}}\)
\(\sqrt{\frac{96p^9}{6p}}\)
- Answer
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\(4p^4\)
\(\sqrt{\frac{108q^{10}}{3q^2}}\)
\(\sqrt{\frac{36}{35}}\)
- Answer
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\(\frac{6}{\sqrt{35}}\)
\(\sqrt{\frac{144}{65}}\)
\(\sqrt{\frac{20}{81}}\)
- Answer
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\(\frac{2\sqrt{5}}{9}\)
\(\sqrt{\frac{211}{96}}\)
\(\sqrt{\frac{96x^7}{121}}\)
- Answer
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\(\frac{4x^3\sqrt{6x}}{11}\)
\(\sqrt{\frac{108y^4}{49}}\)
\(\sqrt{\frac{300m^5}{64}}\)
- Answer
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\(\frac{5m^2\sqrt{3m}}{4}\)
\(\sqrt{\frac{125n^7}{169}}\)
\(\sqrt{\frac{98r^5}{100}}\)
- Answer
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\(\frac{7r^2\sqrt{2r}}{10}\)
\(\sqrt{\frac{180s^{10}}{144}}\)
\(\sqrt{\frac{28q^6}{225}}\)
- Answer
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\(\frac{2q^3\sqrt{7}}{15}\)
\(\sqrt{\frac{150r^3}{256}}\)
\(\sqrt{\frac{75r^9}{s^8}}\)
- Answer
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\(\frac{5r^4\sqrt{3r}}{s^4}\)
\(\sqrt{\frac{72x^5}{y^6}}\)
\(\sqrt{\frac{28p^7}{q^2}}\)
- Answer
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\(\frac{4p^3\sqrt{7p}}{q}\)
\(\sqrt{\frac{45r^3}{s^{10}}}\)
\(\sqrt{\frac{100x^5}{36x^3}}\)
- Answer
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\(\frac{5x}{3}\)
\(\sqrt{\frac{49r^{12}}{16r^6}}\)
\(\sqrt{\frac{121p^5}{81p^2}}\)
- Answer
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\(\frac{11p\sqrt{p}}{9}\)
\(\sqrt{\frac{25r^8}{64r}}\)
\(\sqrt{\frac{32x^{5}y^{3}}{18x^{3}y}}\).
- Answer
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\(\frac{4xy}{3}\)
\(\sqrt{\frac{75r^{6}s^{8}}{48rs^{4}}}\)
\(\sqrt{\frac{27p^{2}q^{10}}{8p^5q^3}}\)
- Answer
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\(\frac{1}{2pq\sqrt{p}}\)
\(\sqrt{\frac{50r^5s^2}{128r^2s^5}}\)
Everyday Math
- 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
- 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
Explain why \(\sqrt{x^4}=x^2\). Then explain why \(\sqrt{x^{16}}=x^8\).
- Answer
-
Answers will vary.
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?