Question

The solution to the equation $4\times 9^{x-1}=3\sqrt{\left(2^{2x+1}\right)}$ is

• A. $3$
• B. $2$
• C. $\frac{3}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C

$\frac{3}{2}$

Explanation for the correct answer:

Equating the given expression:

$$\begin{gathered} 4\times 9^{x-1}=3\sqrt{\left(2^{2x+1}\right)} \\ \Rightarrow \frac{9^{x-1}}{3}=\frac{\sqrt{\left(2^{2x+1}\right)}}{4} \\ \Rightarrow \frac{{\left(3^2\right)}^{x-1}}{3}=\frac{2^{{\displaystyle \frac{2x+1}{2}}}}{2^2} \\ \Rightarrow 3^{2x-2-1}=2^{\frac{\left(2x+1\right)}{2}-2} \\ \Rightarrow 3^{2x-3}=2^{\frac{\left(2x-3\right)}{2}} \\ \Rightarrow {\left(3^{2x-3}\right)}^2=2^{2x-3} \\ \Rightarrow 3^{4x\times 6}=2^{2x-3} \end{gathered}$$

or $2^0\times 3^{4x\times 6}=2^{2x-3}\times 3^0$, On comparing we get,

$4x–6=0\text{and}2x–3=0$

Solving the above two equations we get

$$\begin{gathered} 4x-6=0 \\ \Rightarrow 4x=6 \\ \Rightarrow x=\frac{6}{4} \\ \Rightarrow x=\frac{3}{2} \\ 2x-3=0 \\ \Rightarrow 2x=3 \\ \Rightarrow x=\frac{3}{2} \end{gathered}$$

Therefore, the correct answer is option (C).