Am I allowed to write: $(-2 + 2i)^6 = 2^6(-1 + i)^6$?

Question

I'm caught by a doubt.

Having $z = (-2 + 2i)^6$,
may I write it this way, before searching for a polar representation?

$z = (-2 + 2i)^6 = 2^6(-1 + i)^6$ ?

Answer 1

There is no ambiguity in writing $2^6$, unlike writing something like $2^\frac{1}{6}$ where there are six possible complex solutions. Because of this, the complex number $2^6$ has an argument of zero. Let's say that in polar form, $-1+i = re^{i\theta}$, then you can write your complex number as

$z=(-2+2i)^6=(2(-1+i))^6 = (2re^{i\theta})^6=2^6(re^{i\theta})^6=2^6(-1+i)^6$.

The punchline is that you can indeed take out the factor of two from the brackets before going to polar form under the caveat that (as geetha290krm noted) the power is an integer. If it is not, you will have to take more care.