$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Question

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"?

a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$

b: ${x}^{y^z} = {(x^y)}^z \neq x^{(y^z)}$

Edit: I know a is correct, but what is the reason for this order of operations?

Answer 1

Choice (b) is pointless since you could instead write $x^{yz}$.

To elaborate, sometimes we mean (a) and sometimes we mean (b). We already have a way to denote (b), but no other way to denote (a).

Answer 2

As for you title question,$\;$ x^y^z $\;$is ambiguous.

For the question in your post, as formatted:

$$(x^y)^z = x^{(yz)} \neq x^{\large y^z}$$

So option $a$ is correct.

Answer 3

Of course $x^{y^z} = x^{(y^z)} \ne (x^y)^z$. To outline a proof by contradiction - if you suppose that $x^{y^z} = (x^y)^z$, you can take the counter example of $2^{2^3} = 2^8 = 256 \ne (2^2)^3 = 4^3 = 64$.