Solve 9(x^{-frac{2/3}y)^{3}}{x^{3}(3y^{2/3})^{3}}*(2x^{frac{3/4}y^{1/2})^2}{18(x^3y^-2)^frac{1{2}}}

$$\frac { 9 ( x ^ { - \frac { 2 } { 3 } } y ) ^ { 3 } } { x ^ { 3 } ( 3 y ^ { \frac { 2 } { 3 } } ) ^ { 3 } } \times \frac { ( 2 x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 2 } } ) ^ { 2 } } { 18 ( x ^ { 3 } y ^ { - 2 } ) ^ { \frac { 1 } { 2 } } }$$

Evaluate

$$\frac{2y^{3}}{27x^{5}}$$

Solution Steps

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

$$\frac{9\left(x^{-\frac{2}{3}}\right)^{3}y^{3}}{x^{3}\times \left(3y^{\frac{2}{3}}\right)^{3}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{9x^{-2}y^{3}}{x^{3}\times \left(3y^{\frac{2}{3}}\right)^{3}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{9x^{-2}y^{3}}{x^{3}\times 3^{3}\left(y^{\frac{2}{3}}\right)^{3}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{9x^{-2}y^{3}}{x^{3}\times 3^{3}y^{2}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{9x^{-2}y^{3}}{x^{3}\times 27y^{2}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Cancel out 9y^{2} in both numerator and denominator.

$$\frac{x^{-2}y}{3x^{3}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{2^{2}\left(x^{\frac{3}{4}}\right)^{2}\left(y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{2^{2}x^{\frac{3}{2}}\left(y^{\frac{1}{2}}\right)^{2}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{2^{2}x^{\frac{3}{2}}y^{1}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{y}{3x^{5}}\times \frac{4x^{\frac{3}{2}}y^{1}}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate y to the power of 1 and get y.

$$\frac{y}{3x^{5}}\times \frac{4x^{\frac{3}{2}}y}{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{4x^{\frac{3}{2}}y}{18\left(x^{3}\right)^{\frac{1}{2}}\left(y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{4x^{\frac{3}{2}}y}{18x^{\frac{3}{2}}\left(y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{y}{3x^{5}}\times \frac{4x^{\frac{3}{2}}y}{18x^{\frac{3}{2}}y^{-1}}$$

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

$$\frac{y}{3x^{5}}\times \frac{2y}{9\times \frac{1}{y}}$$

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

$$\frac{y}{3x^{5}}\times \frac{2y^{2}}{9}$$

Multiply \frac{y}{3x^{5}} times \frac{2y^{2}}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{y\times 2y^{2}}{3x^{5}\times 9}$$

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

$$\frac{y^{3}\times 2}{3x^{5}\times 9}$$

Multiply 3 and 9 to get 27.

$$\frac{y^{3}\times 2}{27x^{5}}$$

Differentiate w.r.t. x

$$-\frac{10y^{3}}{27x^{6}}$$