Solve (-8y^3/4/y^3z^6)^-1/3

$$( \frac { - 8 y ^ { 3 / 4 } } { y ^ { 3 } z ^ { 6 } } ) ^ { - 1 / 3 }$$

Evaluate

$$-\frac{y^{\frac{3}{4}}z^{2}}{2}$$

Solution Steps

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

$$\left(\frac{-8}{y^{\frac{9}{4}}z^{6}}\right)^{-\frac{1}{3}}$$

To raise \frac{-8}{y^{\frac{9}{4}}z^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-8\right)^{-\frac{1}{3}}}{\left(y^{\frac{9}{4}}z^{6}\right)^{-\frac{1}{3}}}$$

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

$$\frac{-\frac{1}{2}}{\left(y^{\frac{9}{4}}z^{6}\right)^{-\frac{1}{3}}}$$

Expand \left(y^{\frac{9}{4}}z^{6}\right)^{-\frac{1}{3}}.

$$\frac{-\frac{1}{2}}{\left(y^{\frac{9}{4}}\right)^{-\frac{1}{3}}\left(z^{6}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{9}{4} and -\frac{1}{3} to get -\frac{3}{4}.

$$\frac{-\frac{1}{2}}{y^{-\frac{3}{4}}\left(z^{6}\right)^{-\frac{1}{3}}}$$

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

$$\frac{-\frac{1}{2}}{y^{-\frac{3}{4}}z^{-2}}$$

Express \frac{-\frac{1}{2}}{y^{-\frac{3}{4}}z^{-2}} as a single fraction.

$$\frac{-1}{2y^{-\frac{3}{4}}z^{-2}}$$

Differentiate w.r.t. y

$$-\frac{3z^{2}}{8\sqrt[4]{y}}$$