Solve (frac{x^{-5}y^3z^10}{y^frac{2{3}}})^frac{-3{5}}

$$( \frac { x ^ { - 5 } y ^ { 3 } z ^ { 10 } } { y ^ { \frac { 2 } { 3 } } } ) ^ { \frac { - 3 } { 5 } }$$

Evaluate

$$\frac{x^{3}}{y^{\frac{7}{5}}z^{6}}$$

Solution Steps

Cancel out y^{\frac{2}{3}} in both numerator and denominator.

$$\left(x^{-5}y^{\frac{7}{3}}z^{10}\right)^{\frac{-3}{5}}$$

Fraction \frac{-3}{5} can be rewritten as -\frac{3}{5} by extracting the negative sign.

$$\left(x^{-5}y^{\frac{7}{3}}z^{10}\right)^{-\frac{3}{5}}$$

Expand \left(x^{-5}y^{\frac{7}{3}}z^{10}\right)^{-\frac{3}{5}}.

$$\left(x^{-5}\right)^{-\frac{3}{5}}\left(y^{\frac{7}{3}}\right)^{-\frac{3}{5}}\left(z^{10}\right)^{-\frac{3}{5}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -\frac{3}{5} to get 3.

$$x^{3}\left(y^{\frac{7}{3}}\right)^{-\frac{3}{5}}\left(z^{10}\right)^{-\frac{3}{5}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{7}{3} and -\frac{3}{5} to get -\frac{7}{5}.

$$x^{3}y^{-\frac{7}{5}}\left(z^{10}\right)^{-\frac{3}{5}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -\frac{3}{5} to get -6.

$$x^{3}y^{-\frac{7}{5}}z^{-6}$$

Differentiate w.r.t. x

$$\frac{3x^{2}}{y^{\frac{7}{5}}z^{6}}$$