what is the value of $9^{ \frac{a}{b} } + 16^{ \frac{b}{a} }$ if $3^{a} = 4^{b}$ I have tried to solve this expression but I'm stuck:
$$9^{ \frac{a}{b} } + 16^{ \frac{b}{a} } = 3^{ \frac{2a}{b} } + 4^{ \frac{2b}{a} }$$
and since $3^{a} = 4^{b}$:
$$3^{ \frac{2a}{b} } + 4^{ \frac{2b}{a} } = 4^{ \frac{2b}{b} } + 3^{ \frac{2a}{a} } = 4^{ b } + 3^{ a }$$
Firstly is this right? Secondly how to complete?

You were close. However, you simplified $\frac{2b}b$ into $b$ and $\frac{2a}a$ into $a$ in your final step, which is not right. Correct that mistake, and you should have your solution.
On a less serious note, I would do an extra step here, for clarity:
$$
3^{2a/b} + 4^{2b/a} = (3^a)^{2/b} + (4^b)^{2/a} = 4^{2a/a} + 3^{2b/b}
$$
Also I prefer to not write fractions in exponents unless necessary. I think they're somewhat ugly.
