Solve (27a^frac{2{3}}b^frac{2{3}})^frac{1{3}}

$$( 27 a ^ { \frac { 2 } { 3 } } b ^ { \frac { 2 } { 3 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$3\left(ab\right)^{\frac{2}{9}}$$

Solution Steps

Expand \left(27a^{\frac{2}{3}}b^{\frac{2}{3}}\right)^{\frac{1}{3}}.

$$27^{\frac{1}{3}}\left(a^{\frac{2}{3}}\right)^{\frac{1}{3}}\left(b^{\frac{2}{3}}\right)^{\frac{1}{3}}$$

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

$$27^{\frac{1}{3}}a^{\frac{2}{9}}\left(b^{\frac{2}{3}}\right)^{\frac{1}{3}}$$

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

$$27^{\frac{1}{3}}a^{\frac{2}{9}}b^{\frac{2}{9}}$$

Calculate 27 to the power of \frac{1}{3} and get 3.

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

Differentiate w.r.t. a

$$\frac{2b^{\frac{2}{9}}}{3a^{\frac{7}{9}}}$$