Question

If ${\left(\frac{2}{3}\right)}^x{\left(\frac{3}{2}\right)}^{2x}=\frac{81}{16}\mathrm{then}x=?$
(a) 1
(b) 2
(c) 3
(d) 4

Answer 1

$$\begin{gathered} {\left(\frac{2}{3}\right)}^x{\left(\frac{3}{2}\right)}^{2x}=\frac{81}{16} \\ \Rightarrow {\left(\frac{2}{3}\right)}^x{\left(\frac{2}{3}\right)}^{-2x}=\frac{81}{16} \\ \Rightarrow {\left(\frac{2}{3}\right)}^{x-2x}=\frac{81}{16} \\ \Rightarrow {\left(\frac{2}{3}\right)}^{-x}=\frac{3^4}{2^4} \\ \Rightarrow {\left(\frac{3}{2}\right)}^x=\frac{3^4}{2^4} \\ \Rightarrow x=4 \end{gathered}$$
Hence, the correct answer is option (d).