Question 5: \((ii)\) \(\quad\) \(\large{ \frac{54\times \sqrt[3]{( 27)^{2x}}}{9^{x+1} +216\left( 3^{2x-1}\right)}}\)
Solution:
\[\begin{align*}
&=\frac{54\times \sqrt[3]{( 27)^{2x}}}{9^{x+1} +216\left( 3^{2x-1}\right)}\\
&=\frac{2\times 3^{3} \times \left( 3^{3}\right)^{\frac{2x}{3}}}{3^{2( x+1)} +2^{3} \times 3^{3} \times 3^{2x-1}}\\
&=\frac{2\times 3^{3} \times 3^{2x}}{3^{2x+2} +8\times 3^{3} \times 3^{2x-1}}\\
&=\frac{2\times 3^{2x+3}}{3^{2x+2} +8\times 3^{2x-1+3}}\\
&=\frac{2\times 3^{2x+3}}{3^{2x+2} +8\times 3^{2x+2}}\\
&=\frac{2\times 3^{2x+3}}{3^{2x+2}( 1+8)}\\
&=\frac{2\times 3^{2x+3-2x-2}}{9}\\
&=\frac{2\times 3}{9}\\
&=\frac{2}{3}\\
\end{align*}\]