Why isn't $i$ affected by powers?

Question

When finding roots of complex functions we can write for example:

$$z=2-2i$$

Let's find complex numbers $w$ such that $$w^4 = 2-2i$$

$$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$

This reads: square root of 8 times e to the minus pi over 4 times i (not sure why it's so unclear in the latex).

Now $$w= 8^{1/8} e^{ - \pi / 16 i}$$

How come $i$ doesn't get affected by the power $1/4$? Shouldn't it read $i^{1/4}$?

Answer 1

This is just using the rules of exponents, i.e $(a^b)^c=a^{bc}$

so... if, $\large w^4=\sqrt{8}e^{-\frac{\pi}{4}i}$, then $\large w=(w^4)^{\frac{1}{4}}=(\sqrt{8}e^{-\frac{\pi}{4}i})^{\frac{1}{4}}=8^{\frac{1}{8}}e^{(-\frac{\pi}{4}i)\frac{1}{4}}=8^{\frac{1}{8}}e^{-\frac{\pi}{16}i}$.