Question

If $\frac{9^x\times 3^5\times {\left(27\right)}^3}{3\times {\left(81\right)}^4}=27$, then find the value of $x$.

Answer 1

Finding the value of $x$ by simplifying the given equation.

$\begin{array}{l}\frac{9^x\times 3^5\times {\left(27\right)}^3}{3\times {\left(81\right)}^4}=27\\ \Rightarrow \frac{9^x\times 3^5\times {\left(3^3\right)}^3}{3\times {\left(3^4\right)}^4}={\left(3\right)}^3\\ \Rightarrow 9^x\times 3^5\times {\left(3^3\right)}^3={\left(3\right)}^3\times 3\times {\left(3^4\right)}^4\\ \Rightarrow 9^x\times 3^5\times 3^9=3^3\times 3\times 3^{16}\left[\because {\left(a^m\right)}^n=a^{mn}\right]\\ \Rightarrow 3^{2x}=\frac{3^3\times 3\times 3^{16}}{3^5\times 3^9}\\ \Rightarrow 3^{2x}=3^{3+1+16-5-9}\left[\because a^{p/q}\times a^{r/s}=a^{\frac{p}{q}+\frac{r}{s}}\right]\\ \Rightarrow 3^{2x}=3^6\\ \Rightarrow 2x=6\left[\because a^p=a^q\Rightarrow p=q\right]\\ \Rightarrow x=3\end{array}$

Hence, the value of $x$ is $3$.