Solve (27R^15/8s^6)^-frac{1{3}}

$$( \frac { 27 R ^ { 15 } } { 8 s ^ { 6 } } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$\frac{2s^{2}}{3R^{5}}$$

Solution Steps

To raise \frac{27R^{15}}{8s^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(27R^{15}\right)^{-\frac{1}{3}}}{\left(8s^{6}\right)^{-\frac{1}{3}}}$$

Expand \left(27R^{15}\right)^{-\frac{1}{3}}.

$$\frac{27^{-\frac{1}{3}}\left(R^{15}\right)^{-\frac{1}{3}}}{\left(8s^{6}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 15 and -\frac{1}{3} to get -5.

$$\frac{27^{-\frac{1}{3}}R^{-5}}{\left(8s^{6}\right)^{-\frac{1}{3}}}$$

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

$$\frac{\frac{1}{3}R^{-5}}{\left(8s^{6}\right)^{-\frac{1}{3}}}$$

Expand \left(8s^{6}\right)^{-\frac{1}{3}}.

$$\frac{\frac{1}{3}R^{-5}}{8^{-\frac{1}{3}}\left(s^{6}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{3} to get -2.

$$\frac{\frac{1}{3}R^{-5}}{8^{-\frac{1}{3}}s^{-2}}$$

Calculate 8 to the power of -\frac{1}{3} and get \frac{1}{2}.

$$\frac{\frac{1}{3}R^{-5}}{\frac{1}{2}s^{-2}}$$

Differentiate w.r.t. R

$$-\frac{10s^{2}}{3R^{6}}$$