My teacher gets the following: $(x^{2})^{12-k} = x^{2k-24}$
Where I get the following: $(x^{2})^{12-k} = x^{24-2k}$
I'd like to think of $2(12-k)$ as $2*12 - 2*k$ or $-2k + 24$. Why/how am I wrong?
He did the following: $x^{4k} * {a^{12-k} \over (x^{2})^{12-k}} = x^{4k} * a^{12-k} * x^{2k-24}$
Your first and last sentences contradict each other. I will assume that the last one is correct.
In that case, what the teacher is doing is $$ \frac1{(x^2)^{12-k}}=(x^2)^{k-12}=x^{2k-24} $$