Stewart Precalc 6e Section 3.5: Complex Numbers

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Stewart Precalc 6e Section 3.5: Complex Numbers 0/86
1 Concept Check - Imaginary Unit · Level 1
The imaginary number \(i\) has the property that \(i^2 = \) ?
2 Concept Check - Real and Imaginary Parts · Level 1
For the complex number \(3 + 4 i\), identify the real part and the imaginary part.
3 Concept Check - Complex Conjugate · Level 1
(a) Find the complex conjugate of \(3 + 4 i\), that is, \(\overline{3 + 4 i}\). (b) Calculate \((3 + 4 i) \overline{3 + 4 i}\).
4 Concept Check - Conjugate Root Theorem · Level 1
If \(3 + 4 i\) is a solution of a quadratic equation with real coefficients, what other complex number must also be a solution of the equation?
5 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(-6 + 4 i\).
6 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(\dfrac{-2 - 5 i}{3}\).
7 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(\dfrac{4 + 7 i}{2}\).
8 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(-\dfrac{1}{2}\).
9 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(-\dfrac{2}{3} i\).
10 Real and Imaginary Parts · Level 1
Find the real and imaginary parts of the complex number \(i \sqrt{3}\).
11 Real and Imaginary Parts · Level 2
Find the real and imaginary parts of the complex number \(\sqrt{3} + \sqrt{-4}\).
12 Real and Imaginary Parts · Level 2
Find the real and imaginary parts of the complex number \(2 - \sqrt{-5}\).
13 Arithmetic with Complex Numbers · Level 2
Evaluate \((2 - 5 i) + (3 + 4 i)\) and write the result in the form \(a + b i\).
14 Arithmetic with Complex Numbers · Level 2
Evaluate \((2 + 5 i) + (4 - 6 i)\) and write the result in the form \(a + b i\).
15 Arithmetic with Complex Numbers · Level 2
Evaluate \((-6 + 6 i) + (9 - i)\) and write the result in the form \(a + b i\).
16 Arithmetic with Complex Numbers · Level 2
Evaluate \((3 - 2 i) + \left(-5 - \dfrac{1}{3} i\right)\) and write the result in the form \(a + b i\).
17 Arithmetic with Complex Numbers · Level 2
Evaluate \(\left(7 - \dfrac{1}{2} i\right) - \left(5 + \dfrac{3}{2} i\right)\) and write the result in the form \(a + b i\).
18 Arithmetic with Complex Numbers · Level 2
Evaluate \((-4 + i) - (2 - 5 i)\) and write the result in the form \(a + b i\).
19 Arithmetic with Complex Numbers · Level 2
Evaluate \((-12 + 8 i) - (7 + 4 i)\) and write the result in the form \(a + b i\).
20 Arithmetic with Complex Numbers · Level 2
Evaluate \(6 i - (4 - i)\) and write the result in the form \(a + b i\).
21 Arithmetic with Complex Numbers · Level 2
Evaluate \(4(-1 + 2 i)\) and write the result in the form \(a + b i\).
22 Arithmetic with Complex Numbers · Level 2
Evaluate \(2 i \left(\dfrac{1}{2} - i\right)\) and write the result in the form \(a + b i\).
23 Arithmetic with Complex Numbers · Level 2
Evaluate \((7 - i)(4 + 2 i)\) and write the result in the form \(a + b i\).
24 Arithmetic with Complex Numbers · Level 2
Evaluate \((5 - 3 i)(1 + i)\) and write the result in the form \(a + b i\).
25 Arithmetic with Complex Numbers · Level 2
Evaluate \((3 - 4 i)(5 - 12 i)\) and write the result in the form \(a + b i\).
26 Arithmetic with Complex Numbers · Level 3
Evaluate \(\left(\dfrac{2}{3} + 12 i\right)\left(\dfrac{1}{6} + 24 i\right)\) and write the result in the form \(a + b i\).
27 Arithmetic with Complex Numbers · Level 2
Evaluate \((6 + 5 i)(2 - 3 i)\) and write the result in the form \(a + b i\).
28 Arithmetic with Complex Numbers · Level 2
Evaluate \((-2 + i)(3 - 7 i)\) and write the result in the form \(a + b i\).
29 Powers of i · Level 2
Evaluate \(i^3\) and write the result in the form \(a + b i\).
30 Powers of i · Level 2
Evaluate \((2 i)^4\) and write the result in the form \(a + b i\).
31 Powers of i · Level 3
Evaluate \(i^{1002}\) and write the result in the form \(a + b i\).
32 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{1}{i}\) and write the result in the form \(a + b i\).
33 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{1}{1 + i}\) and write the result in the form \(a + b i\).
34 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{2 - 3 i}{1 - 2 i}\) and write the result in the form \(a + b i\).
35 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{5 - i}{3 + 4 i}\) and write the result in the form \(a + b i\).
36 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{26 + 39 i}{2 - 3 i}\) and write the result in the form \(a + b i\).
37 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{25}{4 - 3 i}\) and write the result in the form \(a + b i\).
38 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{10 i}{1 - 2 i}\) and write the result in the form \(a + b i\).
39 Division of Complex Numbers · Level 3
Evaluate \((2 - 3 i)^{-1}\) and write the result in the form \(a + b i\).
40 Division of Complex Numbers · Level 2
Evaluate \(\dfrac{4 + 6 i}{3}\) and write the result in the form \(a + b i\).
41 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{-3 + 5 i}{15 i}\) and write the result in the form \(a + b i\).
42 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{1}{1 + i} - \dfrac{1}{1 - i}\) and write the result in the form \(a + b i\).
43 Division of Complex Numbers · Level 3
Evaluate \(\dfrac{(1 + 2 i)(3 - i)}{2 + i}\) and write the result in the form \(a + b i\).
44 Radical Expressions with Complex Numbers · Level 2
Evaluate the radical expression \(\sqrt{-25}\) and express the result in the form \(a + b i\).
45 Radical Expressions with Complex Numbers · Level 2
Evaluate the radical expression \(\sqrt{\dfrac{-9}{4}}\) and express the result in the form \(a + b i\).
46 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\sqrt{-3} \sqrt{-12}\) and express the result in the form \(a + b i\).
47 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\sqrt{\dfrac{1}{3}} \sqrt{-27}\) and express the result in the form \(a + b i\).
48 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \((3 - \sqrt{-5})(1 + \sqrt{-1})\) and express the result in the form \(a + b i\).
49 Radical Expressions with Complex Numbers · Level 4
Evaluate the radical expression \((\sqrt{3} - \sqrt{-4})(\sqrt{6} - \sqrt{-8})\) and express the result in the form \(a + b i\).
50 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\dfrac{2 + \sqrt{-8}}{1 + \sqrt{-2}}\) and express the result in the form \(a + b i\).
51 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\dfrac{1 - \sqrt{-1}}{1 + \sqrt{-1}}\) and express the result in the form \(a + b i\).
52 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\dfrac{\sqrt{-36}}{\sqrt{-2} \sqrt{-9}}\) and express the result in the form \(a + b i\).
53 Radical Expressions with Complex Numbers · Level 3
Evaluate the radical expression \(\dfrac{\sqrt{-7} \sqrt{-49}}{\sqrt{28}}\) and express the result in the form \(a + b i\).
54 Equations with Complex Solutions · Level 2
Find all solutions of the equation \(x^2 + 49 = 0\) and express them in the form \(a + b i\).
55 Equations with Complex Solutions · Level 2
Find all solutions of the equation \(9 x^2 + 4 = 0\) and express them in the form \(a + b i\).
56 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 - 4 x + 5 = 0\) and express them in the form \(a + b i\).
57 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 + 2 x + 2 = 0\) and express them in the form \(a + b i\).
58 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 + 2 x + 5 = 0\) and express them in the form \(a + b i\).
59 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 - 6 x + 10 = 0\) and express them in the form \(a + b i\).
60 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 + x + 1 = 0\) and express them in the form \(a + b i\).
61 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 - 3 x + 3 = 0\) and express them in the form \(a + b i\).
62 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(2 x^2 - 2 x + 1 = 0\) and express them in the form \(a + b i\).
63 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(2 x^2 + 3 = 2 x\) and express them in the form \(a + b i\).
64 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(t + 3 + \dfrac{3}{t} = 0\) and express them in the form \(a + b i\).
65 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(z + 4 + \dfrac{12}{z} = 0\) and express them in the form \(a + b i\).
66 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(6 x^2 + 12 x + 7 = 0\) and express them in the form \(a + b i\).
67 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(4 x^2 - 16 x + 19 = 0\) and express them in the form \(a + b i\).
68 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(\dfrac{1}{2} x^2 - x + 5 = 0\) and express them in the form \(a + b i\).
69 Equations with Complex Solutions · Level 3
Find all solutions of the equation \(x^2 + \dfrac{1}{2} x + 1 = 0\) and express them in the form \(a + b i\).
70 Properties of Complex Conjugates · Level 4
Recall that the symbol \(\overline{z}\) represents the complex conjugate of \(z\). If \(z = a + b i\) and \(w = c + d i\), prove that \(\overline{z} + \overline{w} = \overline{z + w}\).
71 Properties of Complex Conjugates · Level 4
If \(z = a + b i\) and \(w = c + d i\), prove that \(\overline{z w} = \overline{z} \cdot \overline{w}\).
72 Properties of Complex Conjugates · Level 4
If \(z = a + b i\), prove that \((\overline{z})^2 = \overline{z^2}\).
73 Properties of Complex Conjugates · Level 3
If \(z = a + b i\), prove that \(\overline{\overline{z}} = z\).
74 Properties of Complex Conjugates · Level 3
If \(z = a + b i\), prove that \(z + \overline{z}\) is a real number.
75 Properties of Complex Conjugates · Level 3
If \(z = a + b i\), prove that \(z - \overline{z}\) is a pure imaginary number.
76 Properties of Complex Conjugates · Level 3
If \(z = a + b i\), prove that \(z \cdot \overline{z}\) is a real number.
77 Properties of Complex Conjugates · Level 4
If \(z = a + b i\), prove that \(z = \overline{z}\) if and only if \(z\) is real.
78 Discovery - Complex Conjugate Roots · Level 4
Suppose that the equation \(a x^2 + b x + c = 0\) has real coefficients and complex (non-real) roots. Why must the roots be complex conjugates of each other? (Hint: Think about how you would find the roots using the Quadratic Formula.)
79 Discovery - Powers of i · Level 4
Calculate the first 12 powers of \(i\), that is, \(i, i^2, i^3, ..., i^{12}\). Do you notice a pattern? Explain how you would calculate any whole number power of \(i\), using the pattern that you have discovered. Use this procedure to calculate \(i^{4446}\).
80 Example - Complex Numbers · Level 1
Identify the real and imaginary parts of each complex number.
(a) \(3 + 4 i\)
(b) \(\dfrac{1}{2} - \dfrac{2}{3} i\)
(c) \(6 i\)
(d) \(-7\)

Enter your answer directly below each part above.

81 Example - Adding, Subtracting, and Multiplying Complex Numbers · Level 2
Express the following in the form \(a + b i\).
(a) \((3 + 5 i) + (4 - 2 i)\)
(b) \((3 + 5 i) - (4 - 2 i)\)
(c) \((3 + 5 i)(4 - 2 i)\)
(d) \(i^{23}\)

Enter your answer directly below each part above.

82 Example - Dividing Complex Numbers · Level 2
Express the following in the form \(a + b i\).
(a) \(\dfrac{3 + 5 i}{1 - 2 i}\)
(b) \(\dfrac{7 + 3 i}{4 i}\)

Enter your answer directly below each part above.

83 Example - Square Roots of Negative Numbers · Level 1
Evaluate the following square roots.
(a) \(\sqrt{-1}\)
(b) \(\sqrt{-16}\)
(c) \(\sqrt{-3}\)

Enter your answer directly below each part above.

84 Example - Using Square Roots of Negative Numbers · Level 3
Evaluate \((\sqrt{12} - \sqrt{-3})(3 + \sqrt{-4})\) and express in the form \(a + b i\).
85 Example - Quadratic Equations with Complex Solutions · Level 2
Solve each equation.
(a) \(x^2 + 9 = 0\)
(b) \(x^2 + 4 x + 5 = 0\)

Enter your answer directly below each part above.

86 Example - Complex Conjugates as Solutions of a Quadratic · Level 3
Show that the solutions of the equation \(4 x^2 - 24 x + 37 = 0\) are complex conjugates of each other.

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