Skip to main content
Mathematics LibreTexts

8.9E: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    30333

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

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

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

    ( \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

    Exercises 1 - 4: Evaluate a Square Root of a Negative Number

    In the following exercises, write each expression in terms of \(i\) and simplify if possible.

    1. a. \(\sqrt{-16}\) b. \(\sqrt{-11}\) c. \(\sqrt{-8}\)
    2. a. \(\sqrt{-121}\) b. \(\sqrt{-1}\) c. \(\sqrt{-20}\)
    3. a. \(\sqrt{-100}\) b. \(\sqrt{-13}\) c. \(\sqrt{-45}\)
    4. a. \(\sqrt{-49}\) b. \(\sqrt{-15}\) c. \(\sqrt{-75}\)
    Answer

    1. a. \(4i\) b. \(i\sqrt{11}\) c. \(2i\sqrt{2}\)

    3. a. \(10i\) b. \(i\sqrt{13}\) c. \(3i\sqrt{5}\)

    ExerciseS 5 - 21: Add or Subtract Complex Numbers

    In the following exercises, add or subtract, putting the answer in \(a + bi\) form.

    5. \(\sqrt{-75}+\sqrt{-48}\)

    6. \(\sqrt{-12}+\sqrt{-75}\)

    7. \(\sqrt{-50}+\sqrt{-18}\)

    8. \(\sqrt{-72}+\sqrt{-8}\)

    9. \((1+3 i)+(7+4 i)\)

    10. \((6+2 i)+(3-4 i)\)

    11. \((8-i)+(6+3 i)\)

    12. \((7-4 i)+(-2-6 i)\)

    13. \((1-4 i)-(3-6 i)\)

    14. \((8-4 i)-(3+7 i)\)

    15. \((6+i)-(-2-4 i)\)

    16. \((-2+5 i)-(-5+6 i)\)

    17. \((5-\sqrt{-36})+(2-\sqrt{-49})\)

    18. \((-3+\sqrt{-64})+(5-\sqrt{-16})\)

    19. \((-7-\sqrt{-50})-(-32-\sqrt{-18})\)

    20. \((-5+\sqrt{-27})-(-4-\sqrt{-48})\)

    Answer

    5. \(0+\left(9\sqrt{3}\right)i\)

    7. \(0+\left(8\sqrt{2}\right)i\)

    9. \(8+7i\)

    11. \(14+2i\)

    13. \(-2+2i\)

    15. \(8+5i\)

    17. \(7-13i\)

    19. \(25-\left(2 \sqrt{2}\right) i\)

    ExerciseS 21 - 28: Multiply Complex Numbers

    In the following exercises, multiply, putting the answer in \(a+bi\) form.

    21. \(4 i(5-3 i)\)

    22. \(2 i(-3+4 i)\)

    23. \(-6 i(-3-2 i)\)

    24. \(-i(6+5 i)\)

    25. \((4+3 i)(-5+6 i)\)

    26. \((-2-5 i)(-4+3 i)\)

    27. \((-3+3 i)(-2-7 i)\)

    28. \((-6-2 i)(-3-5 i)\)

    Answer

    21. \(12+20i\)

    23. \(-12+18i\)

    25. \(-38+9 i\)

    27. \(27+15i\)

    ExerciseS 29 - 32: Multiply Complex Numbers

    In the following exercises, multiply using the Product of Binomial Squares Pattern, putting the answer in \(a+bi\) form.

    29. \((3+4 i)^{2}\)

    30. \((-1+5 i)^{2}\)

    31. \((-2-3 i)^{2}\)

    32. \((-6-5 i)^{2}\)

    Answer

    29. \(-7+24i\)

    31. \(-5-12i\)

    Exercises 33 - 46: Multiply Complex Numbers

    In the following exercises, multiply, putting the answer in \(a+bi\) form.

    33. \(\sqrt{-25} \cdot \sqrt{-36}\)

    34. \(\sqrt{-4} \cdot \sqrt{-16}\)

    35. \(\sqrt{-9} \cdot \sqrt{-100}\)

    36. \(\sqrt{-64} \cdot \sqrt{-9}\)

    37. \((-2-\sqrt{-27})(4-\sqrt{-48})\)

    38. \((5-\sqrt{-12})(-3+\sqrt{-75})\)

    39. \((2+\sqrt{-8})(-4+\sqrt{-18})\)

    40. \((5+\sqrt{-18})(-2-\sqrt{-50})\)

    41. \((2-i)(2+i)\)

    42. \((4-5 i)(4+5 i)\)

    43. \((7-2 i)(7+2 i)\)

    44. \((-3-8 i)(-3+8 i)\)

    Answer

    33. \(30i = 0 + 30i\)

    35. \(-30 = -30 + 0i\)

    37. \(-44+\left(4 \sqrt{3}\right) i\)

    39. \(-20-\left(2 \sqrt{2}\right) i\)

    41. \(5 = 5 + 0i\)

    43. \(53 = 53 + 0i\)

    ExerciseS 45 - 49: Multiply Complex Numbers

    In the following exercises, multiply using the Product of Complex Conjugates Pattern.

    45. \((7-i)(7+i)\)

    46. \((6-5 i)(6+5 i)\)

    47. \((9-2 i)(9+2 i)\)

    48. \((-3-4 i)(-3+4 i)\)

    Answer

    45. \(50\)

    47. \(85\)

    ExerciseS 49 - 60: Divide Complex Numbers

    In the following exercises, divide, putting the answer in \(a+bi\) form.

    49. \(\dfrac{3+4 i}{4-3 i}\)

    50. \(\dfrac{5-2 i}{2+5 i}\)

    51. \(\dfrac{2+i}{3-4 i}\)

    52. \(\dfrac{3-2 i}{6+i}\)

    53. \(\dfrac{3}{2-3 i}\)

    54. \(\dfrac{2}{4-5 i}\)

    55. \(\dfrac{-4}{3-2 i}\)

    56. \(\dfrac{-1}{3+2 i}\)

    57. \(\dfrac{1+4 i}{3 i}\)

    58. \(\dfrac{4+3 i}{7 i}\)

    59. \(\dfrac{-2-3 i}{4 i}\)

    60. \(\dfrac{-3-5 i}{2 i}\)

    Answer

    49. \(i = 0 + i\)

    51. \(\frac{2}{25}+\frac{11}{25} i\)

    53. \(\frac{6}{13}+\frac{9}{13} i\)

    55. \(-\frac{12}{13}-\frac{8}{13} i\)

    57. \(\frac{4}{3}-\frac{1}{3} i\)

    59. \(-\frac{3}{4}+\frac{1}{2} i\)

    ExerciseS 61 - 68: Simplify Powers of \(i\)

    In the following exercises, simplify.

    61. \(i^{41}\)

    62. \(i^{39}\)

    63. \(i^{66}\)

    64. \(i^{48}\)

    65. \(i^{128}\)

    66. \(i^{162}\)

    67. \(i^{137}\)

    68. \(i^{255}\)

    Answer

    61. \(i^{41} = i^{40}\cdot i = \left(i^{4}\right)^{10}\cdot i= i\)

    63. \(i^{66} = i^{64}\cdot i^{2} = \left(i^{4}\right)^{16}\cdot (-1)= -1\)

    65. \(i^{128} = \left(i^{4}\right)^{32} = 1\)

    67. \(i^{137} = i^{136}\cdot i = \left(i^{4}\right)^{34}\cdot i = 1 \cdot i = i\)

    ExerciseS 69 - 72: Writing Exercises

    69. Explain the relationship between real numbers and complex numbers.

    70. Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?
    \(\begin{array}{c}{\sqrt{-7} \cdot \sqrt{-7}} \\ {\sqrt{49}} \\ {7}\end{array}\)

    71. Why is \(\sqrt{-64}=8 i\) but \(\sqrt[3]{-64}=-4\).

    72. Explain how dividing complex numbers is similar to rationalizing a denominator.

    Answer

    69. Answers may vary

    71. Answers may vary

    Self Check

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

    The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “evaluate the square root of a negative number”, “add or subtract complex numbers”, “multiply complex numbers”, “divide complex numbers”, and “simplify powers of i”. The other columns are left blank so the learner can indicate their level of understanding.
    Figure 8.8.15

    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 8.9E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.