[5] If \left(\sqrt{3}\right)^{5} \times 9^{2}=3^{n}\times 3\sqrt{3}, then the value of n is:

For Competitive Exams

If $\left(\sqrt{3}\right)^{5} \times 9^{2}=3^{n}\times 3\sqrt{3}$, then the value of n is:

Explanation:

$\left(\sqrt{3}\right)^{5} \times 9^{2}=3^{n}\times 3\sqrt{3}$

$\Leftrightarrow \left(3^{\dfrac{1}{2}}\right)^{5} \times \left(3^{2}\right)^{2}$ $ = 3^{n}\times3 \times 3^{\dfrac{1}{2}}$

$\Leftrightarrow 3^{\left(\dfrac{1}{2}\times 5\right)}\times 3^{\left(2\times 2\right)}$ $=3^{\left(n+1+\dfrac{1}{2}\right)}$

$\Leftrightarrow3^{\left(\dfrac{5}{2}+4\right)}$ $=3^{\left(n+\dfrac{3}{2}\right)}$

$\Leftrightarrow n+\dfrac{3}{2}=\dfrac{13}{2}$

$\Leftrightarrow n=\left(\dfrac{13}{2}-\dfrac{3}{2}\right)$ $=\dfrac{10}{2}=5$

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