Simplilfying the given radical expression (involving fractional exponents) means removing roots as much as possible, using the laws of exponents (powers)

First apply the rule

\(\sqrt[n]{\frac{x}{y}}=\sqrt[n]{\frac{x}{\sqrt[n]{y}}}\)

Here, \(\displaystyle{x}={11}{a}^{{2}},{y}={64}{b}^{{6}}={\left({4}\right)}^{{3}}{b}^{{6}}={\left({4}{b}^{{2}}\right)}^{{3}}\)

So,

\(\sqrt[3]{\frac{11a^2}{64b^6}}=\sqrt[3]{\frac{{11a^2}}{\sqrt[3]{(4b^2)^3}}}\)

Note: the numerator (radical ) cannot be simplified as none of the factors 11 or a^2 is a cube. On the other hand, the denominator radical can be simplified , as it is a cube.

Now apply the rule

\(\sqrt[n]{z^k}=(z^k)^{\frac{1}{n}}=z^{\frac{k}{n}}\)

Here, \(\displaystyle{z}={4}{b}^{{2}},{k}={3},{n}={3}\)

So,

\(\sqrt[3]{(4b^2)^3}=(4b^2)^{\frac{3}{3}}=4b^2\)

Answer: \(\sqrt[3]{\frac{11a^2}{64b^6}}=\frac{\sqrt[3]{11a^2}}{4b^2}\)