Solve (x^-frac{1{2}}*y^-frac{1{2}}*z^-frac{1{2}})^-frac{2{3}}

$$( x ^ { - \frac { 1 } { 2 } } \cdot y ^ { - \frac { 1 } { 2 } } \cdot z ^ { - \frac { 1 } { 2 } } ) ^ { - \frac { 2 } { 3 } }$$

Evaluate

$$\sqrt[3]{xyz}$$

Solution Steps

Expand \left(x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}\right)^{-\frac{2}{3}}.

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

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

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

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

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

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

$$x^{\frac{1}{3}}y^{\frac{1}{3}}z^{\frac{1}{3}}$$

Differentiate w.r.t. x

$$\frac{\sqrt[3]{yz}}{3x^{\frac{2}{3}}}$$