Question

If ${27}^x=\frac{9}{3^x}$, find $x$.

Answer 1

Step $1$: Write $27$ and $9$ in terms of $3$.

Given : ${27}^x=\frac{9}{3^x}$

we know that,

$27$ can be written as : $27=3^3$

$9$ can be written as : $9=3^2$

So we have, ${\left(3^3\right)}^x=\frac{3^2}{3^x}$

$\Rightarrow 3^{3x}=3^{2-x}$

Step $2$: Compare the power of both sides and solve for $x$.

$$\begin{gathered} 3x=2-x \\ \Rightarrow 4x=2 \\ \Rightarrow x=\frac{2}{4} \\ \Rightarrow \boxed{x=\frac{1}{2}} \end{gathered}$$

Hence, the value of $x$ is $\frac{1}{2}$.