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4.3E: Exercises

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    Practice Makes Perfect

    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. \(\sqrt{-100}\) b. \(\sqrt{-13}\) c. \(\sqrt{-45}\)
    Answer

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

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

    Add or Subtract Complex Numbers

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

    1. \(\sqrt{-75}+\sqrt{-48}\)
    2. \(\sqrt{-50}+\sqrt{-18}\)
    3. \((1+3 i)+(7+4 i)\)
    4. \((8-i)+(6+3 i)\)
    5. \((1-4 i)-(3-6 i)\)
    6. \((6+i)-(-2-4 i)\)
    7. \((5-\sqrt{-36})+(2-\sqrt{-49})\)
    8. \((-7-\sqrt{-50})-(-32-\sqrt{-18})\)
    Answer
    1. \(0+\left(9\sqrt{3}\right)i\)
    2. \(0+\left(8\sqrt{2}\right)i\)
    3. \(8+7i\)
    4. \(14+2i\)
    5. \(-2+2i\)
    6. \(8+5i\)
    7. \(7-13i\)
    8. \(25-\left(2 \sqrt{2}\right) i\)
    Multiply Complex Numbers

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

    1. \(4 i(5-3 i)\)
    2. \(-6 i(-3-2 i)\)
    3. \((4+3 i)(-5+6 i)\)
    4. \((-3+3 i)(-2-7 i)\)
    Answer
    1. \(12+20i\)
    2. \(-12+18i\)
    3. \(-38+9 i\)
    4. \(27+15i\)
    Multiply Complex Numbers

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

    1. \((3+4 i)^{2}\)
    2. \((-1+5 i)^{2}\)
    3. \((-2-3 i)^{2}\)
    4. \((-6-5 i)^{2}\)
    Answer
    1. \(-7+24i\)
    2. \(-24-10i\)
    3. \(-5-12i\)
    4. \(11+60i\)
    Multiply Complex Numbers

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

    1. \((7-i)(7+i)\)
    2. \((6-5 i)(6+5 i)\)
    3. \((9-2 i)(9+2 i)\)
    4. \((-3-4 i)(-3+4 i)\)
    Answer
    1. \(50\)
    2. \(61\)
    3. \(85\)
    4. \(25\)
    Divide Complex Numbers

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

    1. \(\dfrac{3+4 i}{4-3 i}\)
    2. \(\dfrac{2+i}{3-4 i}\)
    3. \(\dfrac{3}{2-3 i}\)
    4. \(\dfrac{-4}{3-2 i}\)
    5. \(\dfrac{1+4 i}{3 i}\)
    6. \(\dfrac{-2-3 i}{4 i}\)
    Answer
    1. \(0 + i\)
    2. \(\frac{2}{25}+\frac{11}{25} i\)
    3. \(\frac{6}{13}+\frac{9}{13} i\)
    4. \(-\frac{12}{13}-\frac{8}{13} i\)
    5. \(\frac{4}{3}-\frac{1}{3} i\)
    6. \(-\frac{3}{4}+\frac{1}{2} i\)

    This page titled 4.3E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.