use DeMoivres theorem to find (-2i)^6

Question

i need to see the step by step solution please help me 

Answer 1

Since you asked to use DeMoivres theorem, I show you the way as below

(-2i)^6

= [2e^(3pi/2 i)]^6, since e^(3pi/2 i) = cos(3pi/2) + i sin(3pi/2) = -i

= 2^6 e^(9pi i)

= 64 e^(pi i), since the period of 2pi

= -64 <==Answer

Answer 2

First, distribute the exponent 6 into the parentheses: 

-2⁶i⁶

We can calculate -2⁶ easily but what about i⁶. Well there exist 4 specific values of iⁿ (n=1,2,3,4) that will help us figure this out. Here are the values in order:

i¹ = √-1

i² = -1

i³ = -√-1 = -i

i⁴ = 1

After i⁵, the values just start repeating from i¹. The way we proceed about figuring out which of these values is i⁶, we first need to take 6 and divide it by 4 BUT we leave the remainder. 

6/4 = 1 R2. So dividing 6 by 4 gives us 1 times and a remainder of 2. This remainder of 2 will be our new exponent for i. Why does this work? Because we're taking out all the i⁴ which are all equal to 1 and in multiplication 1 doesn't change the number it multiplies so we can just toss those out and worry about the remainder, or just remember that the values cycle around every 4th exponent. So now that we have our i, let's put everything together and calculate the answer:

-2⁶i² = 64(-1) (recall from our list of i values, i² = -1) = -64. Answer.