1.4E Exercises
- Page ID
- 152870
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Compute without using a calculator.
- \( 4^2 \)
- \( 3^3\)
- \( \sqrt{25} \)
- \( \sqrt{100} \)
- \( 0^4\)
- \( 1^{13} \)
- \( (-1)^{13} \)
- \( (-1)^{2024} \)
- \( \sqrt{\frac{9}{25}} \)
- \( \sqrt{\frac{1}{4}}\)
- \( \sqrt[3]{-27} \)
- \( \sqrt[3]{-125} \)
- \( 7^{-100} \cdot 7^{101} \)
- \( \frac{3^{12}}{9^{2}} \)
- \( 16^0\)
- \( 1337^0\)
- \( (\sqrt{18})^2 \)
- \( (\sqrt[3]{8})^4 \)
- Answer
-
- 16
- 27
- 5
- 10
- 0
- 1
- -1
- 1
- \( \frac{3}{5} \)
- \( \frac{1}{2}\)
- -3
- -5
- \( 7^{-100+101} = 7^1 = 7 \)
- Since \( 9 = 3^2\), we have \( \frac{3^{12}}{(3^2)^2} = \frac{3^{12}}{3^4} = 3^8 \).
- 1
- 1
- 18
- 16
- Write \( \sqrt{z^7} \) as a fractional power.
- Write \( ( \sqrt[5]{6})^2 \) as a fractional power.
- Write \( 27^{\frac{4}{3}} \) using a radical and simplify.
- Write \( \left( \frac{1}{4} \right)^{\frac{1}{2}} \) using a radical and simplify.
- Answer
-
- \( z^{\frac{7}{2}} \)
- \( 6^{\frac{2}{5}} \)
- \( ( \sqrt[3]{27})^4 = 3^4 = 81 \)
- \( \sqrt{\frac{1}{4}} = \frac{1}{2} \)
Simplify.
1. \( \sqrt{ \frac{4}{9}} - \frac{5}{3} \)
2. \( (2x^2)^6 \)
3. \( \left( \frac{2}{7} \cdot \frac{7}{\sqrt{4}} + 1 \right)^{11} \)
4. \( \left( \dfrac{\sqrt{3} + \sqrt{12}}{6} \right) ^2 \)
5. \( \dfrac{a^7 b^{-3} c}{a^3 b^2 c^{-1} } \)
6. \( (\dfrac{2xy}{3})^2 + \dfrac{4}{3} \)
- Answer
-
1. \( -1 \)
2. \( 2^6x^{12} \)
3. \( 2^{11} \)
4. \( \frac{3}{4} \) (Hint: see the last example in the previous section, part 4.)
5. \( \dfrac{a^4 c^2}{b^5} \) or \( a^4 b^{-5} c^2 \)
6. \( \frac{4x^2y^2+12}{9} \)