\[{\text{Simplify }}\sqrt[n]{{a\sqrt[n]{{a\sqrt[n]{{a\sqrt[n]{{...}}}}}}}},{\text{ }}a > 0.\]
\[x = \sqrt[n]{{a\sqrt[n]{{a\sqrt[n]{{a\sqrt[n]{{...}}}}}}}} \Rightarrow {x^n} = a \cdot x \Leftrightarrow {x^{n - 1}} = a \Leftrightarrow x = \sqrt[{n - 1}]{a}.\]
\[\sqrt[{n - 1}]{a}\]
