How would I simplifiy this fraction with exponents?

Question

How would I simplify this:

$$ \frac{\eta^{k-2}}{\eta_c^k} $$

so that $\eta_c$ can also be raised to the power of $k-2$?

Would there be an $\eta_c^2$ on the top or bottom?

Answer 1

You could simplify it as:

$$\left(\frac{\eta}{\eta_c} \right)^{k-2}\cdot\frac{1}{\eta_c^2}$$

In other words, it would go at the bottom.

Answer 2

It appears that $\eta_c^2$ is going to be on the bottom. Given your problem statement, the solution is baby-simple:

$$ \frac{\eta^{k-2}}{\eta_c^k}=\frac{\eta^{k-2}}{\eta_c^{k-2+2}}=\frac{\eta^{k-2}}{\eta_c^{k-2}\cdot\eta_c^2} $$

If you want to move $\eta_c^2$ to the top, here are the steps to do that:

$$ \frac{\eta^{k-2}}{\eta_c^{k-2}\cdot\eta_c^2}=\frac{\eta^{k-2}}{\eta_c^{k-2}}\cdot\frac{1}{\eta_c^2}=\frac{\eta^{k-2}}{\eta_c^{k-2}}\cdot\frac{1}{\eta_c^{-(-2)}}=\frac{\eta^{k-2}}{\eta_c^{k-2}}\cdot\eta_c^{-2}=\frac{\eta^{k-2}\cdot\eta_c^{-2}}{\eta_c^{k-2}} $$