Solve (x^{frac{4/3}y^{2/3})^{1/2}*(x^{2/3}y^{frac{1}{3}})^{2}}{(x^frac{1{2}})^-8*(y^frac{1{2}})^6}

$$\frac { ( x ^ { \frac { 4 } { 3 } } y ^ { \frac { 2 } { 3 } } ) ^ { \frac { 1 } { 2 } } \cdot ( x ^ { \frac { 2 } { 3 } } y ^ { \frac { 1 } { 3 } } ) ^ { 2 } } { ( x ^ { \frac { 1 } { 2 } } ) ^ { - 8 } \cdot ( y ^ { \frac { 1 } { 2 } } ) ^ { 6 } }$$

Evaluate

$$\frac{x^{6}}{y^{2}}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{4}{3} to get 2.

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

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{2}{3} to get 1.

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

Cancel out y in both numerator and denominator.

$$\frac{x^{2}}{x^{-4}y^{2}}$$

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

$$\frac{x^{6}}{y^{2}}$$

Differentiate w.r.t. x

$$\frac{6x^{5}}{y^{2}}$$