A1. Which expression is equivalent to the given expression $\frac{\sqrt[3]{4 a^{2} \cdot \sqrt[3]{a b} \cdot(a+b)^{0}}}{\left(8 b^{-2} \sqrt{a}\right)^{-\frac{2}{3}}} ; a, b, a+b \neq 0$ ?
(A) $2^{\frac{8}{3}} a^{\frac{10}{9}} b^{-\frac{11}{9}}$
(B) $2^{\frac{2}{3}} a^{\frac{5}{6}} b^{\frac{4}{9}}$
(C) $2^{-3} a^{-\frac{5}{3}} b^{\frac{7}{9}}$
(D) $2^{-\frac{7}{6}} a^{-\frac{10}{3}} b^{\frac{5}{9}}$
(E) $2^{\frac{7}{3}} a^{-2} b^{\frac{1}{6}}$
A1. We write the expression with powers and get $\frac{4^{\frac{1}{3}} \frac{2}{3} a^{\frac{1}{9}} b^{\frac{1}{9}} \cdot 1}{8^{-\frac{2}{3}} b^{\frac{4}{3}} a^{-\frac{1}{3}}}$. We write the numbers 4 and 8 as powers of 2 and express the result without a fraction, obtaining $2^{\frac{2}{3}} \cdot 2^{2} \cdot a^{\frac{2}{3}} \cdot a^{\frac{1}{9}} \cdot a^{\frac{1}{3}} \cdot b^{\frac{1}{9}} \cdot b^{-\frac{4}{3}}$. When we combine the powers, we get the result $2^{\frac{8}{3}} a^{\frac{10}{9}} b^{-\frac{11}{9}}$. The correct answer is $A$.