Solve frac{(x^{3}y^{2})^2}{(x^-1y^-frac{2{3}})^frac{1{4}}}

$$\frac { ( x ^ { 3 } y ^ { 2 } ) ^ { 2 } } { ( x ^ { - 1 } y ^ { - \frac { 2 } { 3 } } ) ^ { \frac { 1 } { 4 } } }$$

Evaluate

$$\sqrt[6]{|y|}y^{4}x^{\frac{25}{4}}$$

Solution Steps

Expand \left(x^{3}y^{2}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Expand \left(x^{-1}y^{-\frac{2}{3}}\right)^{\frac{1}{4}}.

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

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

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

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

$$\frac{x^{6}y^{4}}{x^{-\frac{1}{4}}y^{-\frac{1}{6}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$y^{\frac{25}{6}}x^{\frac{25}{4}}$$

Differentiate w.r.t. x

$$\frac{25\sqrt[6]{|y|}y^{4}x^{\frac{21}{4}}}{4}$$