<s>[INST] Why can I divide a fraction like this? 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 $\frac{\frac{1}{2^{2n}+1}}{2^n}$.
 [/INST] One way of looking at it, is that you are multiplying the numerator and the denominator by $\frac{1}{2^n}$. 
In the numerator you get $\frac{2^n}{2^n}$ which is $1$.
In the denominator you get:
$$\frac{1}{2^n}\cdot (2^{2n} + 1)  =   \frac{2^{2n}}{2^n} + \frac{1}{2^n}  =  2^{2n-n} + \frac{1}{2^n}  =  2^n + \frac{1}{2^n}$$
and there you go:
$$\dfrac{1}{2^n + \dfrac{1}{2^n}}$$
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