9.5E: Exercises
- Page ID
- 30279
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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
Divide Square Roots
In the following exercises, simplify.
\(\frac{\sqrt{27}}{6}\)
- Answer
-
\(\frac{\sqrt{3}}{2}\)
\(\frac{\sqrt{50}}{10}\)
\(\frac{\sqrt{72}}{9}\)
- Answer
-
\(\frac{2\sqrt{2}}{3}\)
\(\frac{\sqrt{243}}{6}\)
\(\frac{2−\sqrt{32}}{8}\)
- Answer
-
\(\frac{1−2\sqrt{2}}{4}\)
\(\frac{3+\sqrt{27}}{9}\)
\(\frac{6+\sqrt{45}}{6}\)
- Answer
-
\(\frac{2+\sqrt{5}}{2}\)
\(\frac{10−\sqrt{200}}{20}\)
\(\frac{\sqrt{80}}{\sqrt{125}}\)
- Answer
-
\(\frac{4}{5}\)
\(\frac{\sqrt{72}}{\sqrt{200}}\)
\(\frac{\sqrt{128}}{\sqrt{72}}\)
- Answer
-
\(\frac{4}{3}\)
\(\frac{\sqrt{48}}{\sqrt{75}}\)
- \(\frac{\sqrt{8x^6}}{2x^2}\)
- \(\frac{\sqrt{200m^5}}{98m}\)
- Answer
-
- \(2x^2\)
- \(\frac{10m^2}{7}\)
- \(\frac{\sqrt{10y^3}}{5y}\)
- \(\frac{\sqrt{108n^7}}{243n^3}\)
\(\frac{\sqrt{75r^3}}{108r}\)
- Answer
-
\(\frac{5r}{6}\)
\(\frac{\sqrt{196q^5}}{484q}\)
\(\frac{\sqrt{108p^{5}q^{2}}}{\sqrt{34p^{3}q^{6}}}\)
- Answer
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\(\frac{3p\sqrt{102}}{17q^2}\)
\(\frac{\sqrt{98rs^{10}}}{\sqrt{2r^{3}s^{4}}}\)
\(\frac{\sqrt{320mn^{5}}}{\sqrt{45m^{7}n^{3}}}\)
- Answer
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\(\frac{8n}{3m^3}\)
\(\frac{\sqrt{810c^{3}d^{7}}}{\sqrt{1000c^{5}d}}\)
\(\frac{\sqrt{98}}{14}\)
- Answer
-
\(\frac{\sqrt{2}}{2}\)
\(\frac{\sqrt{72}}{18}\)
\(\frac{5+\sqrt{125}}{15}\)
- Answer
-
\(\frac{1+\sqrt{3}}{3}\)
\(\frac{6−\sqrt{45}}{12}\)
\(\frac{\sqrt{96}}{\sqrt{150}}\)
- Answer
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\(\frac{4}{5}\)
\(\frac{\sqrt{28}}{\sqrt{63}}\)
\(\frac{\sqrt{26y^7}}{2y}\)
- Answer
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\(y^3\sqrt{13}\)
\(\frac{\sqrt{15x^3}}{\sqrt{3x}}\)
Rationalize a One-Term Denominator
In the following exercises, simplify and rationalize the denominator.
\(\frac{10}{\sqrt{6}}\)
- Answer
-
\(\frac{5\sqrt{6}}{3}\)
\(\frac{8}{\sqrt{3}}\)
\(\frac{6}{\sqrt{7}}\)
- Answer
-
\(\frac{6\sqrt{7}}{7}\)
\(\frac{4}{\sqrt{5}}\)
\(\frac{3}{\sqrt{13}}\)
- Answer
-
\(\frac{3\sqrt{13}}{13}\)
\(\frac{10}{\sqrt{11}}\)
\(\frac{10}{3\sqrt{10}}\)
- Answer
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\(\frac{\sqrt{10}}{3}\)
\(\frac{2}{5\sqrt{2}}\)
\(\frac{4}{9\sqrt{5}}\)
- Answer
-
\(\frac{4\sqrt{5}}{45}\)
\(\frac{9}{2\sqrt{7}}\)
\(−\frac{9}{2\sqrt{3}}\)
- Answer
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\(−\frac{3\sqrt{3}}{2}\)
\(−\frac{8}{3\sqrt{6}}\)
\(\sqrt{\frac{3}{20}}\)
- Answer
-
\(\frac{\sqrt{15}}{10}\)
\(\sqrt{\frac{4}{27}}\)
\(\sqrt{\frac{7}{40}}\)
- Answer
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\(\frac{\sqrt{70}}{20}\)
\(\sqrt{\frac{8}{45}}\)
\(\sqrt{\frac{19}{175}}\)
- Answer
-
\(\frac{\sqrt{133}}{35}\)
\(\sqrt{\frac{17}{192}}\)
Rationalize a Two-Term Denominator
In the following exercises, simplify by rationalizing the denominator.
- \(\frac{3}{3+\sqrt{11}}\)
- \(\frac{8}{1−\sqrt{5}}\)
- Answer
-
- \(\frac{3(3−\sqrt{11})}{−2}\)
- \(−2(1+\sqrt{5})\)
- \(\frac{4}{4+\sqrt{7}}\)
- \(\frac{7}{2−\sqrt{6}}\)
- \(\frac{5}{5+\sqrt{6}}\)
- \(\frac{6}{3−\sqrt{7}}\)
- Answer
-
- \(\frac{5(5−\sqrt{6})}{19}\)
- \(3(3+\sqrt{7})\)
- \(\frac{6}{6+\sqrt{5}}\)
- \(\frac{5}{4−\sqrt{11}}\)
\(\frac{\sqrt{3}}{\sqrt{m}−\sqrt{5}}\)
- Answer
-
\(\frac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m−5}\)
\(\frac{\sqrt{5}}{\sqrt{n}−\sqrt{7}}\)
\(\frac{\sqrt{2}}{\sqrt{x}−\sqrt{6}}\)
- Answer
-
\(\frac{\sqrt{2}(\sqrt{x}+\sqrt{3})}{x−6}\)
\(\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\)
\(\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}−\sqrt{5}}\)
- Answer
-
\(\frac{(\sqrt{r}+\sqrt{5})^2}{r−5}\)
\(\frac{\sqrt{s}−\sqrt{6}}{\sqrt{s}+\sqrt{6}}\)
\(\frac{\sqrt{150x^{2}y^{6}}}{\sqrt{6x^{4}y^{2}}}\)
- Answer
-
\(\frac{5y^2}{x}\)
\(\frac{\sqrt{80p^{3}q}}{\sqrt{5pq^{5}}}\)
\(\frac{15}{\sqrt{5}}\)
- Answer
-
\(3\sqrt{5}\)
\(\frac{3}{5\sqrt{8}}\)
\(\sqrt{\frac{8}{54}}\)
- Answer
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\(\frac{2\sqrt{3}}{9}\)
\(\sqrt{\frac{12}{20}}\)
\(\frac{3}{5+\sqrt{5}}\)
- Answer
-
\(\frac{3(5−\sqrt{5})}{20}\)
\(\frac{20}{4−\sqrt{3}}\)
\(\frac{\sqrt{2}}{\sqrt{x}−\sqrt{3}}\)
- Answer
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\(\frac{\sqrt{2}(\sqrt{x}+\sqrt{3})}{x−3}\)
\(\frac{\sqrt{5}}{\sqrt{y}−\sqrt{7}}\)
\(\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}−\sqrt{8}}\)
- Answer
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\(\frac{(\sqrt{x}+2\sqrt{2})^2}{x−8}\)
\(\frac{\sqrt{m}−\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)
Everyday Math
A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify \(\sqrt{\frac{250}{16}}\) to determine how many seconds it takes for the supply kit to reach the ground.
- Answer
-
\(\frac{5\sqrt{10}}{4}\) seconds
A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify \(\sqrt{\frac{1200}{16}}\) to determine how many seconds it takes for the flare to reach the ocean.
Writing Exercises
- Simplify \(\sqrt{\frac{27}{3}}\) and explain all your steps.
- Simplify \(\sqrt{\frac{27}{5}}\) and explain all your steps.
- Why are the two methods of simplifying square roots different?
- Answer
-
Answers will vary.
- Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator.
- Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator.
- Do you agree that rationalizing the denominator makes calculations easier? Why or why not?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?