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Mathematics LibreTexts

3.1E: Exercises

  • Page ID
    17638
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    Verbal

    Exercise \(\PageIndex{1}\)

    Explain how to add complex numbers.

    Answer

    Add the real parts together and the imaginary parts together. 

    Exercise \(\PageIndex{2}\)

    What is the basic principle in multiplication of complex numbers?

    Exercise \(\PageIndex{3}\)

    Give an example to show the product of two imaginary numbers is not always imaginary.

    Answer

    \(i\) times \(i\) equals –1, which is not imaginary. (answers will vary)

    Exercise \(\PageIndex{4}\)

    What is a characteristic of the plot of a real number in the complex plane?

    Algebraic

    For the following exercises, evaluate the algebraic expressions.

    Exercise \(\PageIndex{5}\)

    If \(f(x)=x^2+x−4\), evaluate \(f(2i)\).

    Answer

    \(−8+2i\)

    Exercise \(\PageIndex{6}\)

    If \(f(x)=x^3−2\), evaluate \(f(i)\).

    Exercise \(\PageIndex{7}\)

    If \(f(x)=x^2+3x+5\),evaluate \(f(2+i)\).

    Answer

    \(14+7i\) 

    Exercise \(\PageIndex{8}\)

    If \(f(x)=2x^2+x−3\), evaluate \(f(2−3i)\).

    Exercise \(\PageIndex{9}\)

    If \(f(x)=\frac{x+1}{2−x}\), evaluate \(f(5i)\).

    Answer

    \(−\frac{23}{29}+\frac{15}{29}i\)

    Exercise \(\PageIndex{10}\)

    If \(f(x)=\frac{1+2x}{x+3}\), evaluate \(f(4i)\).

    Graphical

    For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

    Exercise \(\PageIndex{11}\)

    CNX_Precalc_Figure_03_01_201.jpg

    Answer

    2 real and 0 nonreal  

    Exercise \(\PageIndex{12}\)

    CNX_Precalc_Figure_03_01_202.jpg 

    For the following exercises, plot the complex numbers on the complex plane.

    Exercise \(\PageIndex{13}\)

    \(1−2i\)

    Answer

    CNX_Precalc_Figure_03_01_203.jpg 

    Exercise \(\PageIndex{14}\)

    \(−2+3i\)

    Exercise \(\PageIndex{15}\)

    \(i\)

    Answer

    CNX_Precalc_Figure_03_01_205.jpg

    Exercise \(\PageIndex{16}\)

    \(−3−4i\)

    Numeric

    For the following exercises, perform the indicated operation and express the result as a simplified complex number.

    Exercise \(\PageIndex{17}\)

    \((3+2i)+(5−3i)\)

    Answer

    \(8−i\)

    Exercise \(\PageIndex{18}\)

    \((−2−4i)+(1+6i)\)

    Exercise \(\PageIndex{19}\)

    \((−5+3i)−(6−i)\)

    Answer

    \(−11+4i\)

    Exercise \(\PageIndex{20}\)

    \((2−3i)−(3+2i)\)

    Exercise \(\PageIndex{21}\)

    \((−4+4i)−(−6+9i)\)

    Answer

    \(2−5i\)

    Exercise \(\PageIndex{22}\)

    \((2+3i)(4i)\)

    Exercise \(\PageIndex{23}\)

    \((5−2i)(3i)\)

    Answer

    \(6+15i\)

    Exercise \(\PageIndex{24}\)

    \((6−2i)(5)\)

    Exercise \(\PageIndex{25}\)

    \((−2+4i)(8)\)

    Answer

    \(−16+32i\)

    Exercise \(\PageIndex{26}\)

    \((2+3i)(4−i)\)

    Exercise \(\PageIndex{27}\)

    \((−1+2i)(−2+3i)\)

    Answer

    \(−4−7i\)

    Exercise \(\PageIndex{28}\)

    \((4−2i)(4+2i)\)

    Exercise \(\PageIndex{29}\)

    \((3+4i)(3−4i)\)

    Answer

    25

    Exercise \(\PageIndex{30}\)

    \(\frac{3+4i}{2}\)

    Exercise \(\PageIndex{31}\)

    \(\frac{6−2i}{3}\)

    Answer

    \(2−\frac{2}{3}i\)

    Exercise \(\PageIndex{32}\)

    \(\frac{−5+3i}{2i}\)

    Exercise \(\PageIndex{33}\)

    \(\frac{6+4i}{i}\)

    Answer

    \(4−6i\)

    Exercise \(\PageIndex{34}\)

    \(\frac{2−3i}{4+3i}\)

    Exercise \(\PageIndex{35}\)

    \(\frac{3+4i}{2−i}\)

    Answer

    \(\frac{2}{5}+\frac{11}{5}i\)

    Exercise \(\PageIndex{36}\)

    \(\frac{2+3i}{2−3i}\)

    Exercise \(\PageIndex{37}\)

    \(\sqrt{−9}+3\sqrt{−16}\)

    Answer

    \(15i\)

    Exercise \(\PageIndex{38}\)

    \(−\sqrt{−4}−4\sqrt{−25}\)

    Exercise \(\PageIndex{39}\)

    \(\frac{2+\sqrt{−12}}{2}\)

    Answer

    \(1+i\sqrt{3}\)

    Exercise \(\PageIndex{40}\)

    \(\frac{4+\sqrt{−20}}{2}\)

    Exercise \(\PageIndex{41}\)

    \(i^8\)

    Answer

    \(1\)

    Exercise \(\PageIndex{42}\)

    \(i^{15}\)

    Exercise \(\PageIndex{43}\)

    \(i^{22}\)

    Answer

    \(−1\)

    Technology

    For the following exercises, use a calculator to help answer the questions.

    44. Evaluate \((1+i)^k\) for \(k=4, 8, \) and \(12\).Predict the value if \(k=16\).

     

    45. Evaluate \((1−i)^k\) for \(k=2, 6,\) and \(10\).Predict the value if \(k=14\).

    Answer: 128i

     

    46. Evaluate (1+i)k−(1−i)k for \(k=4\), 8, and 12. Predict the value for \(k=16\).

     

    47. Show that a solution of \(x^6+1=0\) is \(\frac{\sqrt{3}}{2}+\frac{1}{2}i\).

    Answer: \((\frac{\sqrt{3}}{2}+\frac{1}{2}i)^6=−1\)

     

    48. Show that a solution of \(x^8−1=0\) is \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\).

     

    Extensions

    For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

    49. \(\frac{1}{i}+\frac{4}{i^3}\)

    Answer: \(3i\)

     

    50. \(\frac{1}{i^{11}}−\frac{1}{i^{21}}\)

     

    51. \(i^7(1+i^2)\)

    Answer: 0

     

    52. \(i^{−3}+5i^7\)

     

    53. \(\frac{(2+i)(4−2i)}{(1+i)}\)

    Answer: \(5 – 5i\)

     

    54. \(\frac{(1+3i)(2−4i)}{(1+2i)}\)

     

    55. \(\frac{(3+i)^2}{(1+2i)^2}\)

    Answer: \(−2i\)

     

    56. \(\frac{3+2i}{2+i}+(4+3i)\)

     

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

    Answer: \(\frac{9}{2}−\frac{9}{2}i\)

     

    58. \(\frac{3+2i}{1+2i}−\frac{2−3i}{3+i}\)