Solve (frac{x^{6}}{9y^{-4}})^{1/2}frac{(x^{6}y^{3})^-frac{1{3}}}{(x^4y^2)^-frac{1{2}}}

$$( \frac { x ^ { 6 } } { 9 y ^ { - 4 } } ) ^ { \frac { 1 } { 2 } } \frac { ( x ^ { 6 } y ^ { 3 } ) ^ { - \frac { 1 } { 3 } } } { ( x ^ { 4 } y ^ { 2 } ) ^ { - \frac { 1 } { 2 } } }$$

Evaluate

$$\frac{y|y|\left(|x|\right)^{3}}{3}$$

Solution Steps

To raise \frac{x^{6}}{9y^{-4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{\left(x^{6}y^{3}\right)^{-\frac{1}{3}}}{\left(x^{4}y^{2}\right)^{-\frac{1}{2}}}$$

Expand \left(x^{6}y^{3}\right)^{-\frac{1}{3}}.

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{\left(x^{6}\right)^{-\frac{1}{3}}\left(y^{3}\right)^{-\frac{1}{3}}}{\left(x^{4}y^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{x^{-2}\left(y^{3}\right)^{-\frac{1}{3}}}{\left(x^{4}y^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{x^{-2}y^{-1}}{\left(x^{4}y^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{x^{-2}y^{-1}}{\left(x^{4}\right)^{-\frac{1}{2}}\left(y^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{x^{-2}y^{-1}}{x^{-2}\left(y^{2}\right)^{-\frac{1}{2}}}$$

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

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times \frac{x^{-2}y^{-1}}{x^{-2}y^{-1}}$$

Cancel out x^{-2}\times \frac{1}{y} in both numerator and denominator.

$$\frac{\left(x^{6}\right)^{\frac{1}{2}}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times 1$$

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

$$\frac{x^{3}}{\left(9y^{-4}\right)^{\frac{1}{2}}}\times 1$$

Expand \left(9y^{-4}\right)^{\frac{1}{2}}.

$$\frac{x^{3}}{9^{\frac{1}{2}}\left(y^{-4}\right)^{\frac{1}{2}}}\times 1$$

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

$$\frac{x^{3}}{9^{\frac{1}{2}}y^{-2}}\times 1$$

Calculate 9 to the power of \frac{1}{2} and get 3.

$$\frac{x^{3}}{3y^{-2}}\times 1$$

Express \frac{x^{3}}{3y^{-2}}\times 1 as a single fraction.

$$\frac{x^{3}}{3y^{-2}}$$

Differentiate w.r.t. x

$$xy|x||y|$$