Solve (3/5xy^{2})^2:(3/2y)^3-(1/2xy)(-1/5)^2(-2x)-[(-xy)^2:(5y)]

$$( \frac { 3 } { 5 } x y ^ { 2 } ) ^ { 2 } : ( \frac { 3 } { 2 } y ) ^ { 3 } - ( \frac { 1 } { 2 } x y ) ( - \frac { 1 } { 5 } ) ^ { 2 } ( - 2 x ) - [ ( - x y ) ^ { 2 } : ( 5 y ) ]$$

Evaluate

$$-\frac{4yx^{2}}{75}$$

Solution Steps

Multiply x and x to get x^{2}.

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

Expand \left(\frac{3}{5}xy^{2}\right)^{2}.

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

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

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

Calculate \frac{3}{5} to the power of 2 and get \frac{9}{25}.

$$\frac{\frac{9}{25}x^{2}y^{4}}{\left(\frac{3}{2}y\right)^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{\frac{9}{25}x^{2}y^{4}}{\left(\frac{3}{2}\right)^{3}y^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Calculate \frac{3}{2} to the power of 3 and get \frac{27}{8}.

$$\frac{\frac{9}{25}x^{2}y^{4}}{\frac{27}{8}y^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{\frac{9}{25}yx^{2}}{\frac{27}{8}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Divide \frac{9}{25}yx^{2} by \frac{27}{8} by multiplying \frac{9}{25}yx^{2} by the reciprocal of \frac{27}{8}.

$$\frac{\frac{9}{25}yx^{2}\times 8}{27}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{9}{25} and 8 to get \frac{72}{25}.

$$\frac{\frac{72}{25}yx^{2}}{27}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Divide \frac{72}{25}yx^{2} by 27 to get \frac{8}{75}yx^{2}.

$$\frac{8}{75}yx^{2}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Calculate -\frac{1}{5} to the power of 2 and get \frac{1}{25}.

$$\frac{8}{75}yx^{2}-\frac{1}{2}x^{2}y\times \frac{1}{25}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{1}{2} and \frac{1}{25} to get \frac{1}{50}.

$$\frac{8}{75}yx^{2}-\frac{1}{50}x^{2}y\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{1}{50} and -2 to get -\frac{1}{25}.

$$\frac{8}{75}yx^{2}-\left(-\frac{1}{25}x^{2}y\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

The opposite of -\frac{1}{25}x^{2}y is \frac{1}{25}x^{2}y.

$$\frac{8}{75}yx^{2}+\frac{1}{25}x^{2}y-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Combine \frac{8}{75}yx^{2} and \frac{1}{25}x^{2}y to get \frac{11}{75}yx^{2}.

$$\frac{11}{75}yx^{2}-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{11}{75}yx^{2}-\frac{\left(-x\right)^{2}y^{2}}{5y}$$

Calculate -x to the power of 2 and get x^{2}.

$$\frac{11}{75}yx^{2}-\frac{x^{2}y^{2}}{5y}$$

Cancel out y in both numerator and denominator.

$$\frac{11}{75}yx^{2}-\frac{yx^{2}}{5}$$

Combine \frac{11}{75}yx^{2} and -\frac{yx^{2}}{5} to get -\frac{4}{75}yx^{2}.

$$-\frac{4}{75}yx^{2}$$

Expand

$$-\frac{4yx^{2}}{75}$$

Solution Steps

Multiply x and x to get x^{2}.

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

Expand \left(\frac{3}{5}xy^{2}\right)^{2}.

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

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

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

Calculate \frac{3}{5} to the power of 2 and get \frac{9}{25}.

$$\frac{\frac{9}{25}x^{2}y^{4}}{\left(\frac{3}{2}y\right)^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{\frac{9}{25}x^{2}y^{4}}{\left(\frac{3}{2}\right)^{3}y^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Calculate \frac{3}{2} to the power of 3 and get \frac{27}{8}.

$$\frac{\frac{9}{25}x^{2}y^{4}}{\frac{27}{8}y^{3}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{\frac{9}{25}yx^{2}}{\frac{27}{8}}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Divide \frac{9}{25}yx^{2} by \frac{27}{8} by multiplying \frac{9}{25}yx^{2} by the reciprocal of \frac{27}{8}.

$$\frac{\frac{9}{25}yx^{2}\times 8}{27}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{9}{25} and 8 to get \frac{72}{25}.

$$\frac{\frac{72}{25}yx^{2}}{27}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Divide \frac{72}{25}yx^{2} by 27 to get \frac{8}{75}yx^{2}.

$$\frac{8}{75}yx^{2}-\frac{1}{2}x^{2}y\left(-\frac{1}{5}\right)^{2}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Calculate -\frac{1}{5} to the power of 2 and get \frac{1}{25}.

$$\frac{8}{75}yx^{2}-\frac{1}{2}x^{2}y\times \frac{1}{25}\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{1}{2} and \frac{1}{25} to get \frac{1}{50}.

$$\frac{8}{75}yx^{2}-\frac{1}{50}x^{2}y\left(-2\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Multiply \frac{1}{50} and -2 to get -\frac{1}{25}.

$$\frac{8}{75}yx^{2}-\left(-\frac{1}{25}x^{2}y\right)-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

The opposite of -\frac{1}{25}x^{2}y is \frac{1}{25}x^{2}y.

$$\frac{8}{75}yx^{2}+\frac{1}{25}x^{2}y-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

Combine \frac{8}{75}yx^{2} and \frac{1}{25}x^{2}y to get \frac{11}{75}yx^{2}.

$$\frac{11}{75}yx^{2}-\frac{\left(\left(-x\right)y\right)^{2}}{5y}$$

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

$$\frac{11}{75}yx^{2}-\frac{\left(-x\right)^{2}y^{2}}{5y}$$

Calculate -x to the power of 2 and get x^{2}.

$$\frac{11}{75}yx^{2}-\frac{x^{2}y^{2}}{5y}$$

Cancel out y in both numerator and denominator.

$$\frac{11}{75}yx^{2}-\frac{yx^{2}}{5}$$

Combine \frac{11}{75}yx^{2} and -\frac{yx^{2}}{5} to get -\frac{4}{75}yx^{2}.

$$-\frac{4}{75}yx^{2}$$