@the-blade-dancer So sorry for the late response! Basically, the sentence explains why we think that 162 * r^8 = 32/81. It's essentially an educated guess...

As the solution says, the denominator of 32/81 is 3^4. This is the same thing as saying the number has a factor of 3^-4 (because 32/81 = 32 * 1/81 = 32 * (1/3^4) = 32 * 3^-4). On the other hand, 162 has a factor of 3^4 = 81. So to get from 162 to 32/81, thinking purely in terms of the powers of 3, we are trying to get from

3^4 * something to 3^(-4) * something

Well, to do that, we have to divide by 3^8. We just found that the ratio is sqrt(2)/3, so we want to multiply by the ratio 8 times: this gives (sqrt(2))^8 / 3^8 so it's like dividing by 3^8, which is what we want.

I hope that makes sense!