Solve (8a^frac{1{2}}b^frac{1{3}})^frac{2{3}}

$$( 8 a ^ { \frac { 1 } { 2 } } b ^ { \frac { 1 } { 3 } } ) ^ { \frac { 2 } { 3 } }$$

Evaluate

$$4b^{\frac{2}{9}}\sqrt[3]{a}$$

Solution Steps

Expand \left(8a^{\frac{1}{2}}b^{\frac{1}{3}}\right)^{\frac{2}{3}}.

$$8^{\frac{2}{3}}\left(a^{\frac{1}{2}}\right)^{\frac{2}{3}}\left(b^{\frac{1}{3}}\right)^{\frac{2}{3}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and \frac{2}{3} to get \frac{1}{3}.

$$8^{\frac{2}{3}}a^{\frac{1}{3}}\left(b^{\frac{1}{3}}\right)^{\frac{2}{3}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and \frac{2}{3} to get \frac{2}{9}.

$$8^{\frac{2}{3}}a^{\frac{1}{3}}b^{\frac{2}{9}}$$

Calculate 8 to the power of \frac{2}{3} and get 4.

$$4a^{\frac{1}{3}}b^{\frac{2}{9}}$$

Differentiate w.r.t. a

$$\frac{4b^{\frac{2}{9}}}{3a^{\frac{2}{3}}}$$