Powers and Roots of Complex Numbers

Question 1

Review the proof of de Moivre’s theorem. Proof of de Moivre's Theorem [cos(θ) + i sin(θ)]k + 1A= [cos(θ) + i sin(θ)]k ∙ [cos(θ) + i sin(θ)]1B= [cos(kθ) + i sin(kθ)] ∙ [cos(θ) + i sin(θ)]C= cos(kθ)cos(θ) − sin(kθ)sin(θ) + i [sin(kθ)cos(θ) + cos(kθ)sin(θ)]D= ?E= cos[(k + 1)θ] + i sin[(k + 1)θ]

Revolt · Accepted Answer

cos(kθ + θ) + i sin(kθ + θ)

Question 2

Review the graph of z3.

Revolt · Accepted Answer

A

Question 3

If z = 2 + 2i, what is z4?

Revolt · Accepted Answer

64 (cosine (pi) plus i sine (pi))

Question 4

What is the modulus after (–1 + ) gets raised to the 6th power?

Revolt · Accepted Answer

64

Question 5

What is the distance on the unit circle between successive fourth roots of ?

Revolt · Accepted Answer

StartFraction pi Over 2 EndFraction

Question 6

Where are the 4th roots of 16i located?

Revolt · Accepted Answer

on a circle with radius 2

Question 7

Given z = , what is z3?

Revolt · Accepted Answer

64 (cosine ( StartFraction 3pi over 2 EndFraction) plus i sine (StartFraction 3pi over 2 EndFraction))

Question 8

Which number represents a square root of ?

Revolt · Accepted Answer

3 (cosine ( StartFraction pi over 4 EndFraction) plus i sine (StartFraction pi over 4 EndFraction))

Question 9

If z = 1 – i, what is z3?

Revolt · Accepted Answer

–8

Question 10

Which expression represents a cube root of –8?

Revolt · Accepted Answer

2 (cosine ( StartFraction pi over 3 EndFraction) plus i sine (StartFraction pi over 3 EndFraction))

Powers and Roots of Complex Numbers Answers

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