3.1E: Exercises
- Page ID
- 31079
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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 \(\PageIndex{11}\)
- Answer
-
2 real and 0 nonreal
Exercise \(\PageIndex{12}\)
For the following exercises, plot the complex numbers on the complex plane.
Exercise \(\PageIndex{13}\)
\(1−2i\)
- Answer
-
Exercise \(\PageIndex{14}\)
\(−2+3i\)
Exercise \(\PageIndex{15}\)
\(i\)
- Answer
-
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. \(\frac{3+2i}{1+2i}−\frac{2−3i}{3+i}\)