Fractional Exponents Confusion

Question

Let a and b be natural numbers (not including zero). Is it true that will not equal for all possible solutions? For instance, if a=b the would always give an output of x (assuming you don't start shifting around the b terms around when raising it to the a power). However, if a=b and there will be b possible solutions not always giving an output of x. So if this truly is the case does anyone else find that the notation of confusing or am I misssing something?

Answer 1

$$(x^a)^{\frac{1}{b}}=(x^\frac{1}{b})^a;a,b\in \mathbb N^+ $$

For $(x^a)^{\frac{1}{b}}$:

$(x^a)^{\frac{1}{b}}=x^{a*\frac{1}{b}}=x^{\frac{a}{b}}$

Also for $(x^\frac{1}{b})^a$:

$(x^{\frac{1}{b}})^a=x^{\frac{1}{b}*a}=x^{\frac{a}{b}}$

When $a=b$ then $(x^a)^{\frac{1}{b}}=x$ regardless of any real number. Same goes to $(x^\frac{1}{b})^a$, it would still equal $x$.

It should not be possible that $(x^a)^{\frac{1}{b}}\not =(x^\frac{1}{b})^a:a,b\in \mathbb N^+ $unless you can prove me wrong.