<s>[INST] 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?
 [/INST] Here a short way:


*

*$3^a = 4^b \Leftrightarrow 3 = 4^{\frac{b}{a}}$ and $4= 3^{\frac{a}{b}}$

*$\Rightarrow 9^{\frac{a}{b}} = \left(3^{\frac{a}{b}}\right)^2 = 4^2$ and 

*$\Rightarrow 16^{\frac{b}{a}} = \left(4^{\frac{b}{a}}\right)^2 = 3^2$
Now, just sum up.
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