Solve (-64x^3/z^6y^9)^frac{2{3}}

$$( \frac { - 64 x ^ { 3 } } { z ^ { 6 } y ^ { 9 } } ) ^ { \frac { 2 } { 3 } }$$

Evaluate

$$\frac{16x^{2}}{z^{4}y^{6}}$$

Solution Steps

To raise \frac{-64x^{3}}{z^{6}y^{9}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-64x^{3}\right)^{\frac{2}{3}}}{\left(z^{6}y^{9}\right)^{\frac{2}{3}}}$$

Expand \left(-64x^{3}\right)^{\frac{2}{3}}.

$$\frac{\left(-64\right)^{\frac{2}{3}}\left(x^{3}\right)^{\frac{2}{3}}}{\left(z^{6}y^{9}\right)^{\frac{2}{3}}}$$

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

$$\frac{\left(-64\right)^{\frac{2}{3}}x^{2}}{\left(z^{6}y^{9}\right)^{\frac{2}{3}}}$$

Calculate -64 to the power of \frac{2}{3} and get 16.

$$\frac{16x^{2}}{\left(z^{6}y^{9}\right)^{\frac{2}{3}}}$$

Expand \left(z^{6}y^{9}\right)^{\frac{2}{3}}.

$$\frac{16x^{2}}{\left(z^{6}\right)^{\frac{2}{3}}\left(y^{9}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{2}{3} to get 4.

$$\frac{16x^{2}}{z^{4}\left(y^{9}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{2}{3} to get 6.

$$\frac{16x^{2}}{z^{4}y^{6}}$$

Differentiate w.r.t. x

$$\frac{32x}{z^{4}y^{6}}$$