Solve (125r^2s^5/512rs^frac{4{3}})^frac{1{3}}

$$( \frac { 125 r ^ { 2 } s ^ { 5 } } { 512 r s ^ { \frac { 4 } { 3 } } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$\frac{5\sqrt[3]{r}s^{\frac{11}{9}}}{8}$$

Solution Steps

Cancel out rs^{\frac{4}{3}} in both numerator and denominator.

$$\left(\frac{125rs^{\frac{11}{3}}}{512}\right)^{\frac{1}{3}}$$

To raise \frac{125rs^{\frac{11}{3}}}{512} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(125rs^{\frac{11}{3}}\right)^{\frac{1}{3}}}{512^{\frac{1}{3}}}$$

Expand \left(125rs^{\frac{11}{3}}\right)^{\frac{1}{3}}.

$$\frac{125^{\frac{1}{3}}r^{\frac{1}{3}}\left(s^{\frac{11}{3}}\right)^{\frac{1}{3}}}{512^{\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{11}{3} and \frac{1}{3} to get \frac{11}{9}.

$$\frac{125^{\frac{1}{3}}r^{\frac{1}{3}}s^{\frac{11}{9}}}{512^{\frac{1}{3}}}$$

Calculate 125 to the power of \frac{1}{3} and get 5.

$$\frac{5r^{\frac{1}{3}}s^{\frac{11}{9}}}{512^{\frac{1}{3}}}$$

Calculate 512 to the power of \frac{1}{3} and get 8.

$$\frac{5r^{\frac{1}{3}}s^{\frac{11}{9}}}{8}$$

Differentiate w.r.t. r

$$\frac{5s^{\frac{11}{9}}}{24r^{\frac{2}{3}}}$$