Answer of this Question?

Dear Student.

$$\begin{gathered} We have: \\ \sqrt{2}\times \sqrt[3]{3}\times \sqrt[4]{4} \\ =2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}} [\sin ce \sqrt[b]{a}=a^{\frac{1}{b}}] \\ Rewriting the exponents: \\ {2}^{\frac{6}{12}}\times 3^{\frac{4}{12}}\times 4^{\frac{3}{12}} [Since, LCM of 2,3,4 is 12] \\ Now, we get: \\ \sqrt[12]{2^6}\times \sqrt[12]{3^4}\times \sqrt[12]{4^3} [\sin ce a^{\frac{m}{n}}=\sqrt[n]{a^m}] \\ Rearranging the terms: \\ \sqrt[12]{2^6}\times \sqrt[12]{2^6}\times \sqrt[12]{3^4} [\sin ce {\left(a^m\right)}^n=a^{mn}] \\ \sqrt[12]{2^{12}}\times \sqrt[12]{3^4} [\sin ce a^m\times a^n=a^{m+n}] \\ 2\sqrt[12]{81} \\ Hence \left(a\right) is the right option. \end{gathered}$$

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