Answer
$8\sqrt[6] 2$
Work Step by Step
Given: $\frac{64^{\frac{5}{9}}.64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$
Apply the Product of Powers Property: $64^{\frac{5}{9}}.64^{\frac{2}{9}}=64^{{\frac{5}{9}}+{\frac{2}{9}}}=64^{\frac{7}{9}}$
The expression becomes: $\frac{64^{\frac{7}{9}}}{4^{\frac{3}{4}}}=\frac{(2^6)^{\frac{7}{9}}}{(2^2)^{\frac{3}{4}}}=\frac{2^{\frac{42}{9}}}{2^{\frac{3}{2}}}$
Apply the Quotient of Powers Property: $\frac{2^{\frac{42}{9}}}{2^{\frac{3}{2}}}=2^{\frac{42}{9}-{\frac{3}{2}}}=2^{\frac{19}{6}}$
Apply the Product of Powers Property: $2^{\frac{19}{6}}=2^{\frac{18}{6}}.2^{\frac{1}{6}}=2^3.2^{\frac{1}{6}}=8*2^{\frac{1}{6}}$
Hence, the expression becomes $8*2^{\frac{1}{6}}=8\sqrt[6] 2$
