Solving Exponential Equation for x

Simplify the terms: $9^{x - 1/2} = 3^{2x - 1} = \frac{3^{2x}}{3}$ $3^{2x - 3} = \frac{3^{2x}}{27}$ $2^{2x - 2} = \frac{4^x}{4}$

Rearrange base 3 terms to one side and base 2 terms to the other: $3^{2x - 1} + 3^{2x - 3} = 4^x + 2^{2x - 2}$ $3^{2x}\left(\frac{1}{3} + \frac{1}{27}\right) = 4^x\left(1 + \frac{1}{4}\right)$ $3^{2x}\left(\frac{10}{27}\right) = 4^x\left(\frac{5}{4}\right)$ $\frac{3^{2x}}{4^x} = \frac{5}{4} \times \frac{27}{10} = \frac{27}{8}$ $\left(\frac{9}{4}\right)^x = \left(\frac{3}{2}\right)^3$ $\left(\frac{3}{2}\right)^{2x} = \left(\frac{3}{2}\right)^3 \implies 2x = 3 \implies x = \frac{3}{2}$.