Find product of all roots of following equation in complex numbers: $ 6 (4)^ {\frac{2}{x}} -13 (6) ^ {\frac{2}{x}} + 6 (9)^ {\frac{2}{x}} =0$

Question

Find product of all roots of following equation in complex numbers: $$ 6 (4)^ {\frac{2}{x}} -13 (6) ^ {\frac{2}{x}} + 6 (9)^ {\frac{2}{x}} =0$$

First of all this equation has solution because if we draw $f(x)=6 (4)^ {\frac{2}{x}} + 6 (9)^ {\frac{2}{x}} $ and $g(x)= 13 (6) ^ {\frac{2}{x}} $ we find this equation has answer.now we have $6 (2)^ {\frac{1}{x}} -13 (2) ^ {\frac{1}{x}} (3) ^ {\frac{1}{x}} + 6 (3)^ {\frac{1}{x}} =0$ so we have $6 (2)^ {\frac{1}{x}} + 6 (3)^ {\frac{1}{x}} = 13 (2) ^ {\frac{1}{x}} (3) ^ {\frac{1}{x}} $ then $$ \frac{1}{(2)^ {\frac{1}{x}} } + \frac{1}{(3)^ {\frac{1}{x}} } = \frac{13}{6} $$

Answer 1

Hint:

Let $4^{1/x}=a,9^{1/x}=b$

$$0=6a^2-13ab+6b^2=(2a-3b)(3a-2b)$$

$\left(\dfrac23\right)^{2/x}=\dfrac ab=\dfrac23$ or $\dfrac32=\left(\dfrac23\right)^{-1}$

See : Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$