How do you know what power to raise both sides of an equation to if the indexes of the radicals are different?

Step-by-step answer:

When they are equal, they are equal.

Equal quantities raised to the same power are still equal.

It's the eventual solution that may (or may not) present difficulties.

For example, if

P = Q+R

then

P^3 = (Q+R) ^3

where

Q = 2^8

R = 16^2

P = 2^9 = 512

P^3 = 2^27

(Q+R) = (256+256) = 512 = 2^9

(Q+R) ^3 = 512^3 = (2^9) ^3 = 2^27

So they are still equal.