1.6E: Exercises
- Page ID
- 104805
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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
Simplify the following
- \(2^3\)
- \(2^4\)
- \(5^1\)
- \((\frac{1}{4})^{-2}\)
- \(4^{-2}\)
- \(2^0\)
- \(3-2^2\)
- \(2-3^2\)
- \((1+2)^3\)
- \((2-4)^2\)
- \((3+1)^{-2}\)
- \(9(−4)÷(−2)^2\)
- \((−3)^2÷9+(−2)(−7)\)
- \(((2^1)((-3)^3)\)
- Answer
-
- 8
- 16
- 5
- 16
- \(\frac{1}{16}\)
- 1
- 1
- -7
- 27
- 4
- \(\frac{1}{16}\)
- 9
- 15
- -54
In the following exercises, simplify.
- \(\sqrt{64}\)
- \(-\sqrt{81}\)
- \(\sqrt{196}\)
- \(-\sqrt{1}\)
- \(\sqrt{\frac{4}{9}}\)
- \(-\sqrt{0.01}\)
- \(121^\frac{1}{2}\)
- \(9^\frac{1}{2}\)
- \(27^\frac{1}{3}\)
- \(\sqrt[3]{8}\)
- \(\sqrt[5]{32}\)
- Answer
-
- \(8\)
- \(-9\)
- \(14\)
- ((-1\)
- \(\frac{2}{3}\)
- \(-0.1\)
- 11
- 3
- 3
- 2
- 2
In the following exercises, estimate each root by giving the interval of two consecutive whole numbers in which the root lies.
- \(\sqrt{8}\)
- \(\sqrt{27}\)
- \(\sqrt{70}\)
- \(\sqrt[3]{12}\)
- \(\sqrt[3]{32}\)
- \(\sqrt[3]{71}\)
- Answer
-
- \(2<\sqrt{8}<3\)
- \(5<\sqrt{27}<6\)
- \(8<\sqrt{70}<9\)
- \(2<\sqrt[3]{12}<3\)
- \(3<\sqrt[3]{32}<4\)
- \(4<\sqrt[3]{71}<5\)
In the following exercises, simplify using absolute values as necessary.
- \(\sqrt{(-2)^2}\)
- \(\sqrt[3]{(-3)^3}\)
- \(\sqrt[5]{u^{5}}\)
- \(\sqrt[8]{v^{8}}\)
- \(\sqrt[4]{y^{4}}\)
- \(\sqrt[7]{m^{7}}\)
- \(\sqrt{x^{6}}\)
- \(\sqrt{y^{16}}\)
- Answer
-
- 2
- -3
- \(u\)
- \(|v|\)
- \(|y|\)
- \(m\)
- \(|x^{3}|\)
- \(y^{8}\)