1. For the complex number $i$, where $i^2 = -1$, what is the simplified form of $\dfrac{3 + i}{1 - i}$?
2. A complex impedance is given by $-6+7i$. What is the complex conjugate of $-6+7i$ (flip the sign of the imaginary part only)?
3. To combine two complex measurements, subtract one from the other. What is $(8+5i)-(3-9i)$ written in standard form $a+bi$?
4. A complex impedance is modeled as $z=(2+3i)+(7-10i)$. What is $z$ in standard form $a+bi$ after combining real and imaginary parts?
5. Let $x$ be a real number. In the product $(x + 2i)(3 - i)$, what is the real part after multiplying using FOIL and applying $i^2 = -1$?
6. Given $i = \sqrt{-1}$, what is the simplified form of $(4 + 5i) - (1 - 2i)$?
7. For the complex number $i$, where $i^2 = -1$, what is the value of $(3 + 2i) - (5 - 4i)$?
8. A point in the complex plane is represented by $3-4i$. What is the absolute value of $(3-4i)$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify.
9. Which expression is equivalent to $(1 + 6i)(4 - i)$?
10. What is the real part of $ (3 + 5i)(2 - 3i) $?
11. Which expression is equivalent to $(2 + i)(3 - i)$?
12. What is $(7 + 3i) - (2 - 5i)$?
13. Which expression is equivalent to $(3 + 4i)(2 - 5i)$?
14. What is the absolute value of $5 + 12i$?
15. You are asked to subtract two complex quantities and express the result in standard form. What is $(8 + 4i) - (3 - 9i)$?
16. What is the absolute value of $4 + 3i$?
17. When combining two AC phasors, you add their complex representations. What is $(5-12i)+(-9+3i)$ written in standard form $a+bi$?
18. A polynomial step requires multiplying complex numbers and simplifying. What is $(-6+2i)(-1-3i)$? Use FOIL and apply $i^2=-1$ to write $a+bi$.
19. A complex value is given by $-2-11i$, and you need the conjugate to form a real denominator. What is the complex conjugate of $-2-11i$?
20. What is $(4 + 2i) - (1 - 3i)$?
21. You are simplifying the expression $(2-9i)-(11+3i)$ and need the result in standard form $a+bi$. What is the simplified value?
22. For a complex number $-9+12i$, compute its magnitude to determine its distance from the origin. What is the absolute value of $(-9+12i)$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify.
23. What is the absolute value of $7 - 24i$?
24. What is $(1 + 2i)(3 + 4i)$?
25. What is $(3 + 2i)(1 - 4i)$?
26. What is the complex conjugate of $-3 + 7i$?
27. What is $(2 + 3i) - (4 + i)$?
28. What is the absolute value of $6 - 8i$?
29. In simplifying a signal-processing step, you need to multiply two complex numbers and write the result in standard form. What is $(3 + 2i)(4 - 5i)$? (Use FOIL and apply $i^2 = -1$.)
30. To combine two measurements represented as complex numbers, a student adds them. What is $(-10+5i)+(6-8i)$ in standard form $a+bi$?
31. What is the absolute value of $9i$?
32. What is the absolute value of $0 + 5i$?
33. A complex number is given by $12-5i$. What is the complex conjugate of $12-5i$ (flip the sign of the imaginary part only)?
34. A complex number is given by $-9-12i$. What is the absolute value of $(-9-12i)$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify.
35. When combining two AC measurements, you add their complex values and express the result in standard form. What is $(7 - 3i) + (-2 + 11i)$?
36. What are the complex roots of the quadratic function $f(x) = x^2 - 4x + 13$?
37. For the complex number $i$, where $i^2 = -1$, what is the value of $(3 + 2i) - (5 - 4i)$?
38. Let $x$ be a real number. What is $(x+2i)+(3-5i)$ written in standard form $a+bi$?
39. What is $(4 + i)(3 - 2i)$?
40. What is $(2 + 3i)(4 - 5i)$?
41. What is the real part of $(x + 3i)(2 - 4i)$?
42. What is the complex conjugate of $9 + 12i$?
43. What is the absolute value of $-6 + 8i$?
44. What is $(3 + 4i) + (5 - 7i)$?
45. What is $(8 - 5i) + (3 + 2i)$?
46. What is the complex conjugate of $6 - 8i$?
47. What is the complex conjugate of $6 + i$?
48. What is the absolute value of $3 + 4i$?
49. What is the real part of $(x + 4i)(5 + 2i)$?
50. What is $(5 + 6i) - (2 + 3i)$?
51. To simplify an algebraic expression involving $i$, evaluate $(6-2i)-(1+7i)$ in standard form.
52. While combining two phasors, you add the complex numbers $(5 - 9i)$ and $(-3 + 4i)$. What is $(5 - 9i) + (-3 + 4i)$ in standard form $a + bi$?
53. To simplify an expression, evaluate $(2 + 3i) - (7 - 5i)$ and write the result in standard form $a + bi$.
54. To remove imaginary terms from a denominator later, you plan to multiply by a conjugate. What is the complex conjugate of $8 - 11i$?
55. In an AC circuit calculation, a technician multiplies two complex impedances. What is $(3 + 4i)(2 - 5i)$? Use FOIL and apply $i^2 = -1$ to write the result in standard form $a + bi$.
56. A point in the plane is represented by the complex number $6 - 8i$. What is the absolute value of $6 - 8i$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify the radical.
57. In simplifying a polynomial with complex coefficients, you multiply two binomials. What is $(1 - 6i)(-3 - 2i)$? Use FOIL and apply $i^2=-1$ to write the result as $a + bi$.
58. A signal is modeled by the complex number $-7 + 2i$. To compute a reciprocal later, you first need its complex conjugate. What is the complex conjugate of $-7 + 2i$?
59. A complex number is given by $-9 - 12i$. What is the absolute value of $-9 - 12i$? Use $|a+bi| = \sqrt{a^2 + b^2}$ and simplify.
60. A complex impedance is recorded as $-6 + 9i$. To compute a related quantity, you need its complex conjugate (flip the sign of the imaginary part only). What is the complex conjugate of $-6 + 9i$?
61. A signal processing step multiplies two complex gains. Using FOIL and applying $i^2=-1$, what is $(4-3i)(2+5i)$ in standard form $a+bi$?
62. A model uses the product $(x+3i)(2-4i)$, where $x$ is a real number. What is the <u>real part</u> of $(x+3i)(2-4i)$ after using FOIL and applying $i^2=-1$?
63. A point in the complex plane is given by $z = 5 - 12i$. The distance from the origin equals the absolute value of $z$. What is the absolute value of $(5 - 12i)$? (Simplify the radical.)
64. In a computation involving a variable, you multiply two complex numbers and then take the real part of the product. What is the real part of $(x + 3i)(2 - 5i)$? (Expand using FOIL and apply $i^2 = -1$.)
65. A simplification step requires evaluating an expression involving $i$ and writing the result as a real number. What is $i^{10}$? (Use $i^2 = -1$ and the repeating powers of $i$.)
66. Which of the following is equivalent to the expression $\sqrt{-25} + \sqrt{-36} + 2$? (Note: $i = \sqrt{-1}$)
67. What is the absolute value of $-3 + 4i$?
68. What is $(-4 + 4i) + (2 - 6i)$?
69. What is the real part of $(2 + 3i)(4 - i)$?
70. What is the complex conjugate of $7 - 5i$?
71. Which expression is equivalent to $(5 + 6i)(1 - 2i)$?
72. A model uses the product $(x+3i)(2-4i)$, where $x$ is a real number. What is the <u>real part</u> of $(x+3i)(2-4i)$ after using FOIL and applying $i^2=-1$?
73. What is the complex conjugate of $2 - 2i$?
74. What is the complex conjugate of $4i$?
75. What is $(-2 - 3i)(-1 + i)$?
76. What is $(3 + i)(3 - i)$?
77. What is $(-2 + 5i) + (3 - 4i)$?
78. A complex number is written as $-8+3i$. What is the complex conjugate of $-8+3i$ (flip the sign of the imaginary part only)?
79. In an AC circuit calculation, the impedance is modeled by complex numbers. What is $(3+4i)+(5-2i)$ in standard form $a+bi$?
80. A complex expression includes powers of $i$. Evaluate $i^7$ using the fact that $i^2=-1$ and the powers of $i$ repeat every 4.
81. A point in the complex plane is represented by $z=5-12i$. What is the absolute value $|5-12i|$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify.
82. You are given two complex numbers $-4+9i$ and $6+i$. What is $(-4+9i)+(6+i)$ in standard form?
83. In simplifying an expression, a student multiplies two complex numbers. What is $(-4+3i)(-2-7i)$? Use FOIL and apply $i^2=-1$.
84. In an AC circuit calculation, an engineer multiplies two impedances written as complex numbers. What is $(3+4i)(2-5i)$? Use FOIL and apply $i^2=-1$ to write the result in standard form $a+bi$.
85. A student evaluates a product involving a variable real part. What is the real part of $(x+3i)(2-i)$? (Assume $x$ is a real number and use $i^2=-1$.)
86. A point in the complex plane is represented by $8-6i$. What is the absolute value of $(8-6i)$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify the radical.
87. When dividing by a complex number, you often use its conjugate. What is the complex conjugate of $-6+13i$?
88. A student combines two complex-valued signals by addition. What is $(7-3i)+(-2+9i)$ written in standard form $a+bi$?
89. What is $(2 - 3i)(3 + 2i)$?
90. A complex number is given as $10-4i$. You need its conjugate for a rationalization step. What is the complex conjugate of $10-4i$?
91. In simplifying a signal calculation, you need to multiply two complex numbers. What is $(3+4i)(2-5i)$? Use FOIL and apply that $i^2=-1$ to write the result in standard form $a+bi$.
92. To simplify an algebraic expression involving $i$, evaluate and write the result in standard form $a+bi$: $(4+i)(4-i)$. Use FOIL and apply $i^2=-1$.
93. A vector length is computed from the magnitude of a complex number. What is the absolute value of $(12+5i)$? Use $|a+bi|=\sqrt{a^2+b^2}$ and simplify the radical.
94. To simplify a denominator, you compute a product of conjugates. What is $(9+2i)(9-2i)$? Use FOIL and apply $i^2=-1$.
95. In an AC circuit calculation, a voltage is modeled by the complex number $3+4i$ and a current is modeled by $2-5i$. What is $(3+4i)+(2-5i)$ expressed in standard form $a+bi$?
96. What is the absolute value of $-7i$?
97. What is $(3 + 4i) + (5 - 2i)$?
98. What is the real part of $ (5 + 2i)(4 - 3i) $?
99. Which expression is equivalent to $(1 + 3i)(2 + 5i)$?
100. A computation produces the complex number $-7+9i$, and you need its conjugate to eliminate the imaginary part in a product. What is the complex conjugate of $-7+9i$?
101. A complex number is $-12+5i$. What is its complex conjugate, used to form a real product with the original number?
102. To compute a distance in the complex plane, you need the magnitude of a complex number with integer parts. What is the absolute value of $6-8i$? Use $|a+bi| = \sqrt{a^2 + b^2}$ and simplify the radical.
103. For real numbers $x$ and $y$, what is the real part of $(x+yi)(2-3i)$? (Expand using FOIL and apply $i^2=-1$.)
104. A complex number is given by $-9 - 12i$. To find its distance from the origin, compute its absolute value. What is the absolute value of $(-9 - 12i)$? (Simplify the radical.)
105. What is $(1 + 2i)(1 - 2i)$?
106. What is $ (6 + 7i) + (2 - 3i) $?
107. What are the complex roots of the quadratic function $f(x) = x^2 - 4x + 13$?