Post what the question actually asks you for. It seems unlikely that you were asked to solve this, more likely you were asked to show it is true.
What it is doing is simplifying the denominator:
$\displaystyle 3^{-n}+4^{-n}=\dfrac{1}{3^n}+\dfrac{1}{4^n}= \dfrac{4^n+3^n}{3^n4^n}=\dfrac{4^n+3^n}{12^n} $
CB
In detail, they are multiplying by 1 in order to give the fractions a common denominator so that they can add them, making only one fraction:
$\displaystyle \frac{1}{3^n} + \frac{1}{4^n} = \frac{1}{3^n}\cdot 1 + \frac{1}{4^n}\cdot 1 $
$\displaystyle = \frac{1}{3^n}\cdot \left(\frac{4^n}{4^n}\right) + \frac{1}{4^n}\cdot\left(\frac{3^n}{3^n}\right) = \frac{4^n}{3^n\cdot 4^n} + \frac{3^n}{3^n\cdot 4^n}$
$\displaystyle = \frac{4^n + 3^n}{(3\cdot 4)^n} = \frac{4^n + 3^n}{12^n}$
consider also that when you divide fraction one with fraction two, then it's the same as fraction one multiplied by fraction two "upside down".
For instance $\displaystyle 1/2/2$. 2 writtin as a fraction is $\displaystyle 2/1$ so we get
$\displaystyle 1/2*1/2$ (remember that you have to write $\displaystyle 2/1$ 'upside down' so in this case its $\displaystyle 1/2$) and then we get $\displaystyle 1/2*1/2=1/(2*2)=1/4$