Simplify: ${\left(729\right)}^{\frac{1}{6}}$

Given: ${\left(729\right)}^{\frac{1}{6}}$

Prime factorization of $729 = 3\times 3\times 3\times 3\times 3\times 3 = 3^6$

$\therefore {\left(729\right)}^{\frac{1}{6}} = {\left(3^6\right)}^{\frac{1}{6}}$

$\Rightarrow {\left(729\right)}^{\frac{1}{6}} = {\left(3\right)}^{\frac{6}{6}}$   [∵ (a^m)ⁿ = a^mn]

$\Rightarrow {\left(729\right)}^{\frac{1}{6}} = \mathbf{3}$