Simplifying the radicals

Algebra Level 3

The value of \(S\) is an integer \(I\). Enter the value of \(I\).

\[S=\sqrt[5] {41+29\sqrt {2}}-\sqrt[5] {29\sqrt {2}-41}\]

You may not use a calculator.

\(\textbf{Assumptions}\)

Consider \(\sqrt[5] {a}\) to be the principal fifth root of \(a \ \forall a \in \mathbb{R}\).

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