Question

Which of the following is a solution of z^6 = 64i?

Answer 1

Answer:

2(cos135° + isin135°)

Step-by-step explanation:

z^6 = 64i

We need to change i to polar form

cos (90) + isin (90) = i

x^6 = 64 cos (90) + isin (90)

Now we need to take the sixth root of each side

(x^6) ^ 1/6 = ((64)( cos (90) + isin (90)) ^ 1/6

(x^6) ^ 1/6 = ((64) ^ 1/6 * cos (90) + isin (90)) ^ 1/6

(x^6) ^ 1/6 = 2( * cos (90) + isin (90)) ^ 1/6

We we take the roots of the trig functions, we  have 6 roots

360/n  means the roots are 60 degrees are apart

take 90 /6 = 15 degrees

The first root is at 2 (cos (15) + isin (15))

The second root is at 2 (cos (15+60) + isin (15+60))

                           2 (cos (75) + isin (75))

The third root is at 2 (cos (75+60) + isin (75+60))

                           2 (cos (135) + isin (135))

and so on