So what's the difference between $(x^a)^b$ and $x^{a^b}$?

Question

So what's the difference between $(x^a)^b$ and $x^{a^b}$?

Particularly, why does Wolfram Alpha treat them differently?

http://www.wolframalpha.com/input/?i=(x%5E3)%5E2

http://www.wolframalpha.com/input/?i=x%5E3%5E2

Answer 1

$$\left(x^a\right)^b=x^{ab}\neq x^{a^b}$$

So the actual difference is between $\;ab\;$ and $\;a^b\;$ ....quite a difference, isn't it?

Answer 2

$x^{a^b}$ means: calculate $c= a^b$ and then $x^c$

$(x^a)^b$ means: calculate $c=x^a$ and then $c^b$

Answer 3

$(x^a)^b$ = $x^{(a*b)}$

Whereas $x^{(a^b)}$ is x^a^b

For example,

$(x^2)^5$ = $x^{(2*5)}$ = $x^{10}$

$x^{2^5}$ = $x^{(2^5)}$ = $x^{32}$