Question

If ${\left(81\right)}^{\frac{5}{x}}=243$, then $x=.............$

Answer 1

Solve the given equation using the law of radical exponent.

$\begin{array}{l}{\left(81\right)}^{\frac{5}{x}}=243\\ \Rightarrow {\left(3^4\right)}^{\frac{5}{x}}={\left(3\right)}^5\\ \Rightarrow {\left(3\right)}^{4\times \frac{5}{x}}={\left(3\right)}^5\left[\because {\left(a^m\right)}^n=a^{mn}\right]\\ \Rightarrow {\left(3\right)}^{\frac{20}{x}}={\left(3\right)}^5\end{array}$

Equating the power of both sides as both sides have the same base.

$\begin{array}{l}\therefore \frac{20}{x}=5\\ \Rightarrow 5x=20\\ \Rightarrow x=4\end{array}$

Hence, if ${\left(81\right)}^{\frac{5}{x}}=243$, then $x=\mathbf{4}$.