Solve {left({m}^frac{2{3}}{n}^-frac{3{4}}{z}^frac{5{2}}right)}^frac{1{4}}

$${ \left( { m }^{ \frac{ 2 }{ 3 } } { n }^{ - \frac{ 3 }{ 4 } } { z }^{ \frac{ 5 }{ 2 } } \right) }^{ \frac{ 1 }{ 4 } }$$

Evaluate

$$\frac{\sqrt[6]{|m|}z^{\frac{5}{8}}}{n^{\frac{3}{16}}}$$

Solution Steps

Expand \left(m^{\frac{2}{3}}n^{-\frac{3}{4}}z^{\frac{5}{2}}\right)^{\frac{1}{4}}.

$$\left(m^{\frac{2}{3}}\right)^{\frac{1}{4}}\left(n^{-\frac{3}{4}}\right)^{\frac{1}{4}}\left(z^{\frac{5}{2}}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and \frac{1}{4} to get \frac{1}{6}.

$$m^{\frac{1}{6}}\left(n^{-\frac{3}{4}}\right)^{\frac{1}{4}}\left(z^{\frac{5}{2}}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{3}{4} and \frac{1}{4} to get -\frac{3}{16}.

$$m^{\frac{1}{6}}n^{-\frac{3}{16}}\left(z^{\frac{5}{2}}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{2} and \frac{1}{4} to get \frac{5}{8}.

$$m^{\frac{1}{6}}n^{-\frac{3}{16}}z^{\frac{5}{8}}$$

Differentiate w.r.t. m

$$\frac{z^{\frac{5}{8}}sign(m)}{6n^{\frac{3}{16}}\left(|m|\right)^{\frac{5}{6}}}$$