Solve (m^frac{2{3}}*n^frac{3{4}}*z^frac{5{2}})^frac{7{4}}=

$$( m ^ { \frac { 2 } { 3 } } \cdot n ^ { \frac { 3 } { 4 } } \cdot z ^ { \frac { 5 } { 2 } } ) ^ { \frac { 7 } { 4 } } =$$

Evaluate

$$\left(|m|\right)^{\frac{7}{6}}n^{\frac{21}{16}}z^{\frac{35}{8}}$$

Solution Steps

Expand \left(m^{\frac{2}{3}}n^{\frac{3}{4}}z^{\frac{5}{2}}\right)^{\frac{7}{4}}.

$$\left(m^{\frac{2}{3}}\right)^{\frac{7}{4}}\left(n^{\frac{3}{4}}\right)^{\frac{7}{4}}\left(z^{\frac{5}{2}}\right)^{\frac{7}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and \frac{7}{4} to get \frac{7}{6}.

$$m^{\frac{7}{6}}\left(n^{\frac{3}{4}}\right)^{\frac{7}{4}}\left(z^{\frac{5}{2}}\right)^{\frac{7}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{4} and \frac{7}{4} to get \frac{21}{16}.

$$m^{\frac{7}{6}}n^{\frac{21}{16}}\left(z^{\frac{5}{2}}\right)^{\frac{7}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{2} and \frac{7}{4} to get \frac{35}{8}.

$$m^{\frac{7}{6}}n^{\frac{21}{16}}z^{\frac{35}{8}}$$

Differentiate w.r.t. m

$$\frac{7\sqrt[6]{|m|}sign(m)n^{\frac{21}{16}}z^{\frac{35}{8}}}{6}$$