Evaluate 3×9×27×...............×59049.

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{S}=3\times 9\times 27\times .......\times 59049 \\ \mathrm{Now} \mathrm{after} \mathrm{prime} \mathrm{factorisation} \mathrm{of} 59049, \mathrm{we} \mathrm{get}, \\ 59049=3^{10} \\ \Rightarrow \mathrm{S}=3^1\times 3^2\times 3^3\times ....\times 3^{10} \\ =3^{1+2+3+...+10} \\ \mathrm{Now} \\ \mathrm{Sum} \mathrm{of} \mathrm{first} \mathrm{n} \mathrm{natural} \mathrm{numbers}=\frac{\mathrm{n}\left(\mathrm{n}+1\right)}{2} \\ 1+2+3+...+10=\frac{10\left(10+1\right)}{2}=55 \\ \Rightarrow \mathrm{S}=3^{55} \\ \Rightarrow \boxed{\mathbf{3}\mathbf{\times }\mathbf{9}\mathbf{\times }\mathbf{27}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }\mathbf{59049}\mathbf{=}{\mathbf{3}}^{\mathbf{55}}} \end{gathered}$$

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