Solve (-27k^8v^7/8k^2v)^frac{1{3}}

$$( \frac { - 27 k ^ { 8 } v ^ { 7 } } { 8 k ^ { 2 } v } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$-\frac{3\left(kv\right)^{2}}{2}$$

Solution Steps

Cancel out vk^{2} in both numerator and denominator.

$$\left(\frac{-27k^{6}v^{6}}{8}\right)^{\frac{1}{3}}$$

To raise \frac{-27k^{6}v^{6}}{8} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-27k^{6}v^{6}\right)^{\frac{1}{3}}}{8^{\frac{1}{3}}}$$

Expand \left(-27k^{6}v^{6}\right)^{\frac{1}{3}}.

$$\frac{\left(-27\right)^{\frac{1}{3}}\left(k^{6}\right)^{\frac{1}{3}}\left(v^{6}\right)^{\frac{1}{3}}}{8^{\frac{1}{3}}}$$

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

$$\frac{\left(-27\right)^{\frac{1}{3}}k^{2}\left(v^{6}\right)^{\frac{1}{3}}}{8^{\frac{1}{3}}}$$

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

$$\frac{\left(-27\right)^{\frac{1}{3}}k^{2}v^{2}}{8^{\frac{1}{3}}}$$

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

$$\frac{-3k^{2}v^{2}}{8^{\frac{1}{3}}}$$

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

$$\frac{-3k^{2}v^{2}}{2}$$

Differentiate w.r.t. v

$$-3vk^{2}$$