Skip to main content
Mathematics LibreTexts

4.5E: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    58457

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Practice Makes Perfect

    Exercise A: add and subtract radical expressions

    In the following exercises, simplify. Assume all variables are greater than or equal to zero so that absolute values are not needed.

    1. a. \(8 \sqrt{2}-5 \sqrt{2}\quad\) b. \(5 \sqrt[3]{m}+2 \sqrt[3]{m}\quad\) c. \(8 \sqrt[4]{m}-2 \sqrt[4]{n}\)

    2. a. \(7 \sqrt{2}-3 \sqrt{2}\quad\) b. \(7 \sqrt[3]{p}+2 \sqrt[3]{p}\quad\) c. \(5 \sqrt[3]{x}-3 \sqrt[3]{x}\)

    3. a. \(3 \sqrt{5}+6 \sqrt{5}\quad\) b. \(9 \sqrt[3]{a}+3 \sqrt[3]{a}\quad\) c. \(5 \sqrt[4]{2 z}+\sqrt[4]{2 z}\)

    4. a. \(4 \sqrt{5}+8 \sqrt{5} \quad \) b. \(\sqrt[3]{m}-4 \sqrt[3]{m} \quad \) c. \(\sqrt{n}+3 \sqrt{n}\)

    5. a. \(3 \sqrt{2 a}-4 \sqrt{2 a}+5 \sqrt{2 a} \quad \) b. \(5 \sqrt[4]{3 a b}-3 \sqrt[4]{3 a b}-2 \sqrt[4]{3 a b}\)

    6. a. \(\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b} \quad \) b. \(8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}\)

    7. a. \(8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c} \quad \) b. \(2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}\)

    8. a. \(3 \sqrt{5 d}+8 \sqrt{5 d}-11 \sqrt{5 d} \quad \) b. \(11 \sqrt[3]{2 r s}-9 \sqrt[3]{2 r s}+3 \sqrt[3]{2 r s}\)

    9. a. \(\sqrt{27}-\sqrt{75} \quad \) b. \(\sqrt[3]{40}-\sqrt[3]{320} \quad \) c. \(\frac{1}{2} \sqrt[4]{32}+\frac{2}{3} \sqrt[4]{162}\)

    10. a. \(\sqrt{72}-\sqrt{98} \quad \) b. \(\sqrt[3]{24}+\sqrt[3]{81} \quad \) c. \(\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}\)

    11. a. \(\sqrt{48}+\sqrt{27} \quad \) b. \(\sqrt[3]{54}+\sqrt[3]{128} \quad \) c. \(6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}\)

    12. a. \(\sqrt{45}+\sqrt{80} \quad \) b. \(\sqrt[3]{81}-\sqrt[3]{192} \quad \) c. \(\frac{5}{2} \sqrt[4]{80}+\frac{7}{3} \sqrt[4]{405}\)

    13. a. \(\sqrt{72 a^{5}}-\sqrt{50 a^{5}} \quad \) b. \(9 \sqrt[4]{80 p^{4}}-6 \sqrt[4]{405 p^{4}}\)

    14. a. \(\sqrt{48 b^{5}}-\sqrt{75 b^{5}} \quad \) b. \(8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}\)

    15. a. \(\sqrt{80 c^{7}}-\sqrt{20 c^{7}} \quad \) b. \(2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}\)

    16. a. \(\sqrt{96 d^{9}}-\sqrt{24 d^{9}} \quad \) b. \(5 \sqrt[4]{243 s^{6}}+2 \sqrt[4]{3 s^{6}}\)

    17. \(3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}}\)

    18. \(3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}\)
    Answer

    1. a. \(3 \sqrt{2}\) b. \(7 \sqrt[3]{m}\) c. \(6 \sqrt[4]{m}\)

    3. a. \(9 \sqrt{5}\) b. \(12 \sqrt[3]{a}\) c. \(6 \sqrt[4]{2 z}\)

    5. a. \(4 \sqrt{2 a}\) b. \(0\)

    7. a. \( \sqrt{3c}\) b. \(\sqrt[3]{4 p q}\)

    9. a. \(-2 \sqrt{3}\) b. \(-2 \sqrt[3]{5}\) c. \(3 \sqrt[4]{2}\)

    11. a. \(7 \sqrt{3}\) b. \(7 \sqrt[3]{2}\) c. \(3 \sqrt[4]{5}\)

    13. a. \(a^{2} \sqrt{2 a}\) b. \(0\)

    15. a. \(2 c^{3} \sqrt{5 c}\) b. \(14 r^{2} \sqrt[4]{2 r^{2}}\)

    17. \(4 y \sqrt{2}\)

    Exercise B: multiply radical expressions

    In the following exercises, simplify.

      1. \((-2 \sqrt{3})(3 \sqrt{18})\)

      2. \((8 \sqrt[3]{4})(-4 \sqrt[3]{18})\)



        3.  
      3.  

      1. \((-4 \sqrt{5})(5 \sqrt{10})\)

      2. \((-2 \sqrt[3]{9})(7 \sqrt[3]{9})\)



        3.  
      3.  

      1. \((5 \sqrt{6})(-\sqrt{12})\)

      2. \((-2 \sqrt[4]{18})(-\sqrt[4]{9})\)



        3.  
      3.  

      1. \((-2 \sqrt{7})(-2 \sqrt{14})\)

      2. \((-3 \sqrt[4]{8})(-5 \sqrt[4]{6})\)



        3.  
      3.  

      1. \(\left(4 \sqrt{12 z^{3}}\right)(3 \sqrt{9 z})\)

      2. \(\left(5 \sqrt[3]{3 x^{3}}\right)\left(3 \sqrt[3]{18 x^{3}}\right)\)



        3.  
      3.  

      1. \(\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)\)

      2. \(\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)\)



        3.  
      3.  

      1. \(\left(-2 \sqrt{7 z^{3}}\right)\left(3 \sqrt{14 z^{8}}\right)\)

      2. \(\left(2 \sqrt[4]{8 y^{2}}\right)\left(-2 \sqrt[4]{12 y^{3}}\right)\)



        3.  
      3.  

      1. \(\left(4 \sqrt{2 k^{5}}\right)\left(-3 \sqrt{32 k^{6}}\right)\)

      2. \(\left(-\sqrt[4]{6 b^{3}}\right)\left(3 \sqrt[4]{8 b^{3}}\right)\)



        3.  
      3.  
    Answer

    1.

    1. \(-18 \sqrt{6}\)

    2. \(-64 \sqrt[3]{9}\)



      3.  
    3.  

    3.

    1. \(-30 \sqrt{2}\)

    2. \(6 \sqrt[4]{2}\)



      3.  
    3.  

    5.

    1. \(72 z^{2} \sqrt{3}\)

    2. \(45 x^{2} \sqrt[3]{2}\)



      3.  
    3.  

    7.

    1. \(-42 z^{5} \sqrt{2 z}\)

    2. \(-8 y \sqrt[4]{6 y}\)
    Exercise C: use polynomial multiplication to multiply radical expressions

    In the following exercises, multiply.

      1. \(\sqrt{7}(5+2 \sqrt{7})\)

      2. \(\sqrt[3]{6}(4+\sqrt[3]{18})\)



        3.  
      3.  

      1. \(\sqrt{11}(8+4 \sqrt{11})\)

      2. \(\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})\)



        3.  
      3.  

      1. \(\sqrt{11}(-3+4 \sqrt{11})\)

      2. \(\sqrt[4]{3}(\sqrt[4]{54}+\sqrt[4]{18})\)



        3.  
      3.  

      1. \(\sqrt{2}(-5+9 \sqrt{2})\)

      2. \(\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})\)



        3.  
      3.  

    5. \((7+\sqrt{3})(9-\sqrt{3})\)

    6. \((8-\sqrt{2})(3+\sqrt{2})\)

      1. \((9-3 \sqrt{2})(6+4 \sqrt{2})\)

      2. \((\sqrt[3]{x}-3)(\sqrt[3]{x}+1)\)



        3.  
      3.  

      1. \((3-2 \sqrt{7})(5-4 \sqrt{7})\)

      2. \((\sqrt[3]{x}-5)(\sqrt[3]{x}-3)\)



        3.  
      3.  

      1. \((1+3 \sqrt{10})(5-2 \sqrt{10})\)

      2. \((2 \sqrt[3]{x}+6)(\sqrt[3]{x}+1)\)



        3.  
      3.  

      1. \((7-2 \sqrt{5})(4+9 \sqrt{5})\)

      2. \((3 \sqrt[3]{x}+2)(\sqrt[3]{x}-2)\)



        3.  
      3.  

    11. \((\sqrt{3}+\sqrt{10})(\sqrt{3}+2 \sqrt{10})\)

    12. \((\sqrt{11}+\sqrt{5})(\sqrt{11}+6 \sqrt{5})\)

    13. \((2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})\)

    14. \((4 \sqrt{6}+7 \sqrt{13})(8 \sqrt{6}-3 \sqrt{13})\)

      1. \((3+\sqrt{5})^{2}\)

      2. \((2-5 \sqrt{3})^{2}\)



        3.  
      3.  

      1. \((4+\sqrt{11})^{2}\)

      2. \((3-2 \sqrt{5})^{2}\)



        3.  
      3.  

      1. \((9-\sqrt{6})^{2}\)

      2. \((10+3 \sqrt{7})^{2}\)



        3.  
      3.  

      1. \((5-\sqrt{10})^{2}\)

      2. \((8+3 \sqrt{2})^{2}\)



        3.  
      3.  

    19. \((4+\sqrt{2})(4-\sqrt{2})\)

    20. \((7+\sqrt{10})(7-\sqrt{10})\)

    21. \((4+9 \sqrt{3})(4-9 \sqrt{3})\)

    22. \((1+8 \sqrt{2})(1-8 \sqrt{2})\)

    23. \((12-5 \sqrt{5})(12+5 \sqrt{5})\)

    24. \((9-4 \sqrt{3})(9+4 \sqrt{3})\)

    25. \((\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)\)

    26. \((\sqrt[3]{4 x}+3)(\sqrt[3]{4 x}-3)\)
    Answer

    1.

    1. \(14+5 \sqrt{7}\)

    2. \(4 \sqrt[3]{6}+3 \sqrt[3]{4}\)



      3.  
    3.  

    3.

    1. \(44-3 \sqrt{11}\)

    2. \(3 \sqrt[4]{2}+\sqrt[4]{54}\)



      3.  
    3.  

    5. \(60+2 \sqrt{3}\)

    7.

    1. \(30+18 \sqrt{2}\)

    2. \(\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-3\)



      3.  
    3.  

    9.

    1. \(-54+13 \sqrt{10}\)

    2. \(2 \sqrt[3]{x^{2}}+8 \sqrt[3]{x}+6\)



      3.  
    3.  

    11. \(23+3 \sqrt{30}\)

    13. \(-439-2 \sqrt{77}\)

    15.

    1. \(14+6 \sqrt{5}\)

    2. \(79-20 \sqrt{3}\)

    17.

    1. \(87-18 \sqrt{6}\)

    2. \(163+60 \sqrt{7}\)

    19. \(14\)

    21. \(-227\)

    23. \(19\)

    25. \(\sqrt[3]{9 x^{2}}-4\)

    Exercise D: mixed practice
    1. \(\frac{2}{3} \sqrt{27}+\frac{3}{4} \sqrt{48}\)

    2. \(\sqrt{175 k^{4}}-\sqrt{63 k^{4}}\)

    3. \(\frac{5}{6} \sqrt{162}+\frac{3}{16} \sqrt{128}\)

    4. \(\sqrt[3]{24}+\sqrt[3]{ 81}\)

    5. \(\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}\)

    6. \(8 \sqrt[4]{13}-4 \sqrt[4]{13}-3 \sqrt[4]{13}\)

    7. \(5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}\)

    8. \(\sqrt{80 a^{5}}-\sqrt{45 a^{5}}\)

    9. \(\frac{3}{5} \sqrt{75}-\frac{1}{4} \sqrt{48}\)

    10. \(21 \sqrt[3]{9}-2 \sqrt[3]{9}\)

    11. \(8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}\)

    12. \(11 \sqrt{11}-10 \sqrt{11}\)

    13. \(\sqrt{3} \cdot \sqrt{21}\)

    14. \((4 \sqrt{6})(-\sqrt{18})\)

    15. \((7 \sqrt[3]{4})(-3 \sqrt[3]{18})\)

    16. \(\left(4 \sqrt{12 x^{5}}\right)\left(2 \sqrt{6 x^{3}}\right)\)

    17. \((\sqrt{29})^{2}\)

    18. \((-4 \sqrt{17})(-3 \sqrt{17})\)

    19. \((-4+\sqrt{17})(-3+\sqrt{17})\)

    20. \(\left(3 \sqrt[4]{8 a^{2}}\right)\left(\sqrt[4]{12 a^{3}}\right)\)

    21. \((6-3 \sqrt{2})^{2}\)

    22. \(\sqrt{3}(4-3 \sqrt{3})\)

    23. \(\sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18})\)

    24. \((\sqrt{6}+\sqrt{3})(\sqrt{6}+6 \sqrt{3})\)
    Answer

    1. \(5\sqrt{3}\)

    3. \(9\sqrt{2}\)

    5. \(-\sqrt[4]{5}\)

    7. \(10 c^{2} \sqrt{3}-9 c^{3} \sqrt{3}\)

    9. \(2 \sqrt{3}\)

    11. \(17 q^{2}\)

    13. \(3 \sqrt{7}\)

    15. \(-42 \sqrt[3]{9}\)

    17. \(29\)

    19. \(29-7 \sqrt{17}\)

    21. \(72-36 \sqrt{2}\)

    23. \(6+3 \sqrt[3]{2}\)

    Exercise E: writing exercises
    1. Explain when a radical expression is in simplest form.
    2. Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.
      1. Explain why \((-\sqrt{n})^{2}\) is always non-negative, for \(n \geq 0\).
      2. Explain why \(-(\sqrt{n})^{2}\) is always non-positive, for \(n \geq 0\).
    4. Use the binomial square pattern to simplify \((3+\sqrt{2})^{2}\). Explain all your steps.
    Answer

    1. Answers will vary

    3. Answers will vary

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “add and subtract radical expressions.”, “ multiply radical expressions”, and “use polynomial multiplication to multiply radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.
    Figure 8.4.14

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

    This page titled 4.5E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Author
      OpenStax
      License
      CC BY
      Transcluded
      yes
    2. Tags
      1. source[1]-math-30329