I know about $(x^y)^z$ = $x^{yz}$ but what about $x^{(y^z)}$? Are there any rules for this?
Let's consider the power of power rule: $(x^y)^z$ = $x^{yz}$:
$(2^3)^4$ = $8^4$ = 4096
$2^{3\times 4}$ = $2^{12}$ = 4096
But same can't work for $x^{(y^z)}$. Is there really no rule for this?
I'm not good with calculus so I didn't check those discussions really. If there is such a rule, please explain it.
It all depends on what you consider "a rule". There are many ways to calculate with things, from which many rules stem. One not unimportant rule would be $$ x^{(y^z)} = (x^y)^{(y^{z-1})} = ((\ldots(x^y)^y)\ldots)^y $$ This will matter if you for example want to calculate such a power in some group. Suppose we have some element $x$ of some finite group (let’s say an elliptic curve) and suitable integers $y,z$ and we want to calculate $x^{(y^z)}$. Then $y^z$ might be so large that it does not fit into memory. But instead of calculating $y^z$ we can turn the whole thing into taking $z$ times the $y$th power.