Suppose I have a fraction: $$\frac{2^n}{2^{2n}+1}$$
I can simplify it to become: $$\frac{1}{2^{n}+\frac{1}{2^n}}$$
Now obviously, this is just dividing both the numerator and the denominator of the fraction by $2^n.$ My question is why I can do this. Can anyone explain the algebra behind this division to me?
EDIT: I tried thinking about the initial fraction as $2^n \cdot \frac{1}{2^{2n}+1}$ and the division operation as $\frac{2^n \cdot \frac{1}{2^{2n}+1}}{2^n}$, but I couldn't get anywhere with attempting to compute $\frac{\frac{1}{2^{2n}+1}}{2^n}$.