<s>[INST] Find solution set of $\dfrac{8^x+27^x}{12^x+18^x}=\dfrac{14}{12}$ What I've done is factoring it.
$$\dfrac{2^{3x}+3^{3x}}{2^{2x}\cdot 3^{x}+3^{2x}\cdot{2^{x}}}=\dfrac{7}{2\cdot 3}$$
This looks like it can be factored more but it doesn't work from my attempts.
 [/INST] Well you've reached a good point:
$$\dfrac{2^{3x}+3^{3x}}{2^{2x}\cdot 3^{x}+3^{2x}\cdot{2^{x}}}$$
$$=\dfrac{2^{3x}+3^{3x}}{2^{x}\cdot 3^{x}(2^x+3^{x})}$$
We can let $2^x=a$ and $3^x=b$ and get
$$\frac{a^3+b^3}{ab(a+b)}=\frac{a^2-ab+b^2}{ab}=\frac{a}{b}-1+\frac{b}{a}$$
Now let $z=\frac{a}{b}$, we get
$$z-1+\frac{1}{z}=\frac{7}{6} \iff z^2-\frac{13}{6}z+1=0 \implies z=\frac{2}{3},\frac{3}{2} \implies x=\pm 1$$
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