What do you call this arithmetical property and how do you prove it?

Question

I think I can prove this property only when exponents are integers. But here is my example:

$a(x-b)^{1/2} = (a^2x-a^2b)^{1/2}$

This type of foiling is weird and I wish to see a proof of this.

Namely, for integer exponents I thought I could do this:

$a(x-b)^3 = a^{2/3}*(x-b)*a^{2/3}*(x-b)*a^{2/3}*(x-b) = (a^{2/3}x-a^{2/3}b)^3$

Thanks!

Answer 1

As $\displaystyle (a^x)^y=a^{xy}$ where $x,y, a(>0)$ are real numbers

$$a^m(x-b)^n=\{a^{\frac mn}(x-b)\}^n=(a^{\frac mn}x-a^{\frac mn}b)^n$$