Let's solve this problem step by step. We are given the expression \(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\) and asked to simplify it.

First, let's simplify the innermost square root:

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

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

Now, let's simplify the next square root:

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

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

Finally, let's simplify the outermost square root:

= \(\sqrt{169r^2}\)

= 13r