Question

Use DeMoivre's Theorem to find the indicated power of the following complex number.
(-3 + 3i)⁴

My work

r = √18 and tan θ = -1 so θ = 135° because the complex number is in Quadrant II.

√18 [cos(135°) + i sin(135°)]
(√18)⁴ [cos(4 × 135°) + i sin(4 × 135°)]
324[cos(540°) + i sin(540°)]
324(-1 + i × 0)
324(-1)
-324

Answer 1

I just need to know if I am correct

Answer 2

Hmm I took a very different approach, got the same result though.
Yay good job, looks correct! :)

Answer 3

https://www.wolframalpha.com/input/?i=%28-3%2B3i%29%5E4
Just in case you need to verify ^

Answer 4

If you dont mind, can you show me how you did it?

Answer 5

either way thanks a lot :)

Answer 6

My method is a little strange.
I factor a 3 out to start.\[\large\rm 3^4(-1+i)^4\]I recognize that the magnitude of the real and imaginary parts are the same, that only happens at the pi/4 angles.
So we need to turn these into sqrt(2)/2's.
So I'll factor a sqrt(2) out of each term.\[\large\rm 3^4(\sqrt{2})^4\left(-\frac{\sqrt2}{2}+\frac{\sqrt2}{2}i\right)^4\]And as you said, this is in quadrant 2, so it's 3pi/4.\[\large\rm 3^4(\sqrt{2})^4\left(\cos\frac{3\pi}{4}+i \sin\frac{3\pi}{4}\right)^4\]Then, De'Moivre gives us,\[\large\rm 3^4(\sqrt{2})^4\left(\cos3\pi+i \sin3\pi\right)\]Sine is 0 at the 3pi angle, cosine -1,\[\large\rm 3^4(\sqrt{2})^4\left(-1+0\right)\]And then just simplify a little further to get the same result.

I know, I know, my method is a little goofy :) But I like it lol

Answer 7

Hahaha thanks a lot man, whatever gets the job done :P