Question

If $8^x=\frac{64}{2^x}$ , then $x=$ _____

Answer 1

Solve for $x$

Given, $8^x=\frac{64}{2^x}$

$\Rightarrow 8^x\times 2^x=64$

$$\begin{gathered} \Rightarrow 2^{3x}\times 2^x=64\left[\because 8=2^3 \\ \because 64=2^6\right] \end{gathered}$$

$\Rightarrow 2^{3x+x}=2^6\left[\because x^a\times x^b=x^{a+b}\right]$

$\Rightarrow 2^{4x}=2^6$

$$\begin{gathered} \Rightarrow 4x=6\left[\because a^x=a^y \\ \Rightarrow x=y\right] \end{gathered}$$

$\Rightarrow x=\frac{6}{4}=\frac{3}{2}$

Hence, the value of $x$ is $\frac{3}{2}$.