\(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\)

Simplifying from the innermost radical and progressing outwards we have the given expression

\(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\) = \(\sqrt{160r^2 + \sqrt{71r^4 + 10r^4}}\)

= \(\sqrt{160r^2 + \sqrt{81r^4}}\)

\(\sqrt{160r^2 + 9r^2}\) = \(\sqrt{169r^2}\)

= 13r