Answers PrecalculusPowers and Roots of Complex Numbers

Powers and Roots of Complex Numbers — Unit test Answers

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Review the four points on the complex plane.

Which graph represents points on the polar curve r = 2 + 5sin(θ)?

On a polar coordinate plane, points are at (0, 0), (2, 0), (negative 2, pi), (3, StartFraction 3 pi Over 2 EndFraction), (7, StartFraction 3 pi Over 2 EndFraction).

On a polar coordinate plane, points are at (0, 0), (2, 0), (negative 2, pi), (3, StartFraction pi Over 2 EndFraction), (7, StartFraction pi Over 2 EndFraction).

On a polar coordinate plane, points are at (0, 0), (2, StartFraction pi Over 2 EndFraction), (2, StartFraction 3 pi Over 2 EndFraction), (3, pi), (7, pi).

On a polar coordinate plane, points are at (0, 0), (2, StartFraction pi Over 2 EndFraction), (2, StartFraction 3 pi Over 2 EndFraction), (3, 0), (7, 0).

Given z1 = and , what is the value of z1 – z2?

cos(0) + isin(0)

cos(0) – isin(0)

cos() + isin()

cos() – isin()

Let z = and .What is the product of zw?

40 (cosine (StartFraction pi Over 16 EndFraction) + I sine (StartFraction pi Over 16 EndFraction) )

76 (cosine (StartFraction pi Over 16 EndFraction) + I sine (StartFraction pi Over 16 EndFraction) )

40 (cosine (StartFraction 3 pi Over 16 EndFraction) + I sine (StartFraction 3 pi Over 16 EndFraction) )

76 (cosine (StartFraction 3 pi Over 16 EndFraction) + I sine (StartFraction 3 pi Over 16 EndFraction) )

What is the value of ?

–3 + 6i

3 + 6i

5 + 5i

5 – 5i

What is the modulus and argument after gets raised to the 6th power?

modulus = ; argument =

modulus = 27; argument =

modulus = 729; argument =

Given w = and , what is w – z expressed in polar form?

StartRoot 2 EndRoot (cosine (StartFRaction pi Over 4 EndFraction) + I sine (StartFraction pi Over 4 EndFraction) )

StartRoot 2 EndRoot (cosine (StartFraction 3 pi Over 4 EndFraction) + I sine (StartFraction 3 pi Over 4 EndFraction) )

StartRoot 2 EndRoot (cosine (StartFraction 5 pi Over 4 EndFraction) + I sine (StartFraction 5 pi Over 4 EndFraction) )

StartRoot 2 EndRoot (cosine (StartFraction 7 pi Over 4 EndFraction) + I sine (StartFraction 7 pi Over 4 EndFraction) )

What is the additive identity of the complex number 14 + 5i?

14 – 5i

–14 – 5i

Which equation represents the rectangular form of r = 4cos(θ)?

(x – 2)2 + y2 = 4

(x + 2)2 + y2 = 4

(x – 2)2 + y2 = 0

(x + 2)2 + y2 = 0

Review the four points shown on the polar coordinate system.

point A

point B

point C

point D

Which expression represents a 5th root of –i?

Option

What is the midpoint of a segment in the complex plane with endpoints at 6 – 2i and –4 + 6i?

1 + 2i

2 + i

2 + 4i

5 + 4i

When asked to find the distance between the complex points 3 + 8i and 7 + 6i, Zach showed the work below. He asked his friend Zöe to find his mistake, and she correctly told him that he should have

subtracted the real and imaginary portions of the point in the same order.

left the i out of the computations when finding distance in the complex plane.

remembered that –4 squared should be –16, not 16.

simplified to , not to .

For the complex number ,what is the polar form?

z = five-halves (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction)

z = five-fourths (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction)

z = five-halves (cosine (StartFraction 11 pi Over 6 EndFraction) + I sine (StartFraction 11 pi Over 6 EndFraction)

z = five-fourths (cosine (StartFraction 11 pi Over 6 EndFraction) + I sine (StartFraction 11 pi Over 6 EndFraction)

Consider w1 = 4 + 2i and w2 = –1 – 3i. Which graph represents the sum w1 + w2?

On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line goes from (0, 0) to point (3, negative 5).

On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line goes from (0, 0) to point (3, negative 1).

On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line goes from (0, 0) to point (5, 5).

On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. A line goes from (0, 0) to point (5, 1).

In the complex plane, the rectangular coordinates (x, y) represent a complex number. Which statement explains why polar coordinates (r, θ) represent the same complex number?

r is equivalent to and θ is equivalent to .