<s>[INST] Help in solving exponential equation Solve the following equation:
$$\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}$$
All I managed to do is rewrite the given equation in a simpler form:
$$\frac{4^x}{6^x + 9^x} + \frac{9^x}{6^x + 4^x} = \frac{7}{6}$$
I don't know what should be done next.
Thank you in advance!
 [/INST] Hint:
$$\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}$$
$$6\cdot8^x-7\cdot12^x-7\cdot18^x+6\cdot27^x=0$$
$$6\cdot2^{3x}-7\cdot2^{2x}\cdot3^x-7\cdot2^x\cdot3^{2x}+6\cdot3^{3x}=0$$
Let $\left(\frac23\right)^x=t$
Then
$$6\cdot t^{3}-7\cdot t^2-7\cdot t+6=0$$
$$t\in \{-1;\frac23;\frac32\}$$
$$x=\pm1$$
</s>