AnswersPrecalculusPowers and Roots of Complex Numbers

Powers and Roots of Complex Numbers — Unit test Answers

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1
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Rectangular coordinates and polar coordinates both represent the same complex number z.Which set of equations demonstrates why both sets of coordinates represent the same number?

Question illustration
A
r = StartRoot 3 EndRoot + StartRoot 3 EndRoot = StartRoot 6 EndRoot and theta = inverse tangent of (1) = StartFraction pi over 4 EndFraction
Option A
B
r = StartRoot 3 EndRoot + StartRoot 3 EndRoot = StartRoot 6 EndRoot and theta = inverse tangent of (StartRoot 3 EndRoot) = StartFraction pi over 4 EndFraction
Option B
C
r = StartStartRoot (StartRoot 3 EndRoot) squared + (StartRoot 3 EndRoot) squared EndEndRoot = StartRoot 6 EndRoot and theta = inverse tangent of (1) = StartFraction pi Over 4 EndFraction
Option C
D
r = StartStartRoot (StartRoot 3 EndRoot) squared + (StartRoot 3 EndRoot) squared EndEndRoot = StartRoot 6 EndRoot and theta = inverse tangent of (StartRoot 3 EndRoot) = StartFraction pi Over 4 EndFraction
Option D
2
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Which values of x and y would make the following expression represent a real number? (6 + 3i)(x + yi)

A
x = 6, y = 0
B
x = –3, y = 0
C
x = 6, y = –3
D
x = 0, y = –3
3

Which property of multiplication is shown below? If x = 3 + 4i and y = 5 + 6i, x × y = y × x.

A
commutative property
B
identity property
C
distributive property
D
associative property
4

Which equation represents the rectangular form of r = 8?

A
x2 + y2 = 16
B
x2 + y2 = 64
C
StartRoot x squared + y squared EndRoot = 16
Option C
D
StartRoot x squared + y squared EndRoot = 64
Option D
5

Let z = 19 + i and w = 4 + 10i.What is the first step in determining zw?

A
Replace i2 with –1.
B
Multiply z by the conjugate of w.
C
Apply the distributive property or FOIL.
D
Combine the corresponding real and imaginary parts.
6

Review the graph of complex number z.

Question illustration
A
(cos(135°) + isin(135°))
Option A
B
(cos(135°) + isin(135°))
Option B
C
(cos(225°) + isin(225°))
Option C
D
(cos(225°) + isin(225°))
Option D
7

If z = 1 + i, what is z5?

Question illustration
A
16 + 16i
Option A
B
–16 + 16i
Option B
C
16 – 16i
Option C
D
–16 – 16i
Option D
8

Consider z = 3 + 3i. What happens to the modulus and argument when z is raised to the 4th power?

Question illustration
A
The modulus increases by a factor of 216, and the argument increases by .
Option A
B
The modulus increases by a factor of 216, and the argument increases by .
Option B
C
The modulus increases by a factor of 1,296, and the argument increases by .
Option C
D
The modulus increases by a factor of 1,296, and the argument increases by .
Option D
9

A point of the form 25 +bi is 37 units from 13 – 31i. What is the value of b?

A
4
B
6.6
C
36.6
D
40.6
10

Which region on the unit circle contains a cube root of – 8 – 8i?

A
in quadrant I
B
in quadrant II
C
on the imaginary axis
D
on the real axis
11

For two complex numbers z and w, which diagram shows the geometric representation of zw?

A
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line with length r 1 goes to point z, a line with length r 1 + r 2 goes to point z w, and a line with length r 2 goes to point w. The angle to line w is theta 2, z is theta 1, z w is theta 1 minus theta 2.
Option A
B
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line with length r 1 goes to point z, a line with length r 1 r 2 goes to point z w, and a line with length r 2 goes to point w. The angle to line w is theta 2, z is theta 1, z w is theta 1 minus theta 2.
Option B
C
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line with length r 1 + r 2 goes to point z w, a line with length r 1 goes to point z, and a line with length r 2 goes to point w. The angle to line w is theta 2, z is theta 1, z w is theta 1 + theta 2.
Option C
D
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line with length r 1 r 2 goes to point z w, a line with length r 1 goes to point z, and a line with length r 2 goes to point w. The angle to line w is theta 2, z is theta 1, z w is theta 1 + theta 2.
Option D
12

Which phrase describes how to divide a complex number z by r(cos θ + i sin θ)?

A
Scale by a factor of and rotate clockwise by θ.
Option A
B
Scale by a factor of r and rotate clockwise by θ.
C
Scale by a factor of and rotate counterclockwise by θ.
Option C
D
Scale by a factor of r and rotate counterclockwise by θ.
13

What is the quotient of ?

Question illustration
A
45 (cosine (StartFraction pi Over 3 EndFraction) + I sine (StartFraction pi Over 3 EndFraction) )
Option A
B
88 (cosine (StartFraction pi Over 3 EndFraction) + I sine (StartFraction pi Over 3 EndFraction) )
Option B
C
45 (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction) )
Option C
D
88 (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction) )
Option D
14

Which graph represents points on the polar curve r = –4sin(3θ)?

A
On a polar coordinate plane, points are at (4, StartFraction 2 pi Over 4 EndFraction), (4, 0), and (4, StartFraction 4 pi Over 3 EndFraction).
Option A
B
On a polar coordinate plane, points are at (4, pi), (4, StartFraction pi Over 3 EndFraction), and (4, StartFraction 5 pi Over 3 EndFraction).
Option B
C
On a polar coordinate plane, points are at (4, StartFraction pi Over 6 EndFraction), (4, StartFraction 5 pi Over 6 EndFraction), and (4, StartFraction 3 pi Over 2 EndFraction).
Option C
D
On a polar coordinate plane, point are at (4, StartFraction pi Over 2 EndFraction), (4, StartFraction 11 pi Over 6 EndFraction), (4, StartFraction 7 pi Over 6 EndFraction).
Option D
15

Which expression is equivalent to (3 + 5i)2?

A
–16 + 30i
B
6 + 34i
C
–16 – 34i
D
16 + 30i

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