Is $n^{n\cdot n}$ equal to $(n^n)^n$?

Question

I have a simple question, is $n^{n\cdot n}=(n^n)^n$? I believe it does, because, for example $(n^2)^2 = n^{2\cdot 2}$. Also $1^1 = 1\cdot1, 2^2 = 2\cdot2, 3^3 = 3\cdot3, 4^4 = 4\cdot 4$, so I suppose $n^n = n\cdot n$, am i right?

Answer 1

A law of exponents says that $$ (a^b)^c = a^{bc}. $$ In other words, iterated exponents multiply. Now let $a=b=c=n$ to get $(n^n)^n=n^{n^2}$. Your final claim of $n^n = n^2$ is wrong (let $n=3$) and doesn't follow from the previous (correct) fact.