What is the correct subsitution

Question

I have $a^{n-k}, \; a=x^2$. When I substitute, do I get $x^{2(n-k)}$ or $x^{2^{n-k}}$? I'm confused about which one is correct.

Answer 1

The correct substitution goes as follows:

$$a^{n-k}=(x^2)^{n-k}=x^{2(n-k)}$$

The key here is that $(a^b)^c=a^{bc}$. Notably, exponents aren't associative, so $a^{(b^c)}\neq (a^b)^c$.

Answer 2

If $n=k$, then

$$a^{n-k}=a^0=1$$

This is regardless of what $a$ is. Thus, it shouldn't matter what happens when we let $a=x^2$:

$$x^{2(n-k)}=x^{2(0)}=x^0=1\quad\color{green}{\checkmark\text{right}}$$

$$x^{2^{n-k}}=x^{2^0}=x^1\ne1\implies\color{red}{\text{wrong}}$$