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

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

Evaluate

$$-10$$

Solution Steps

Expand \left(-\frac{1}{5}xy\right)^{2}.

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

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

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

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

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

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

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

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

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

Combine \frac{1}{2}xy and -\frac{1}{5}xy to get \frac{3}{10}xy.

$$\frac{-\frac{2}{25}x^{3}y^{3}+\frac{3}{10}xy\left(\frac{1}{3}x^{2}y^{2}+\frac{1}{5}x^{2}y^{2}\right)}{\left(-\frac{1}{5}xy\right)^{3}}$$

Combine \frac{1}{3}x^{2}y^{2} and \frac{1}{5}x^{2}y^{2} to get \frac{8}{15}x^{2}y^{2}.

$$\frac{-\frac{2}{25}x^{3}y^{3}+\frac{3}{10}xy\times \frac{8}{15}x^{2}y^{2}}{\left(-\frac{1}{5}xy\right)^{3}}$$

Multiply \frac{3}{10} and \frac{8}{15} to get \frac{4}{25}.

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

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

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

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

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

Combine -\frac{2}{25}x^{3}y^{3} and \frac{4}{25}x^{3}y^{3} to get \frac{2}{25}x^{3}y^{3}.

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

Expand \left(-\frac{1}{5}xy\right)^{3}.

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

Calculate -\frac{1}{5} to the power of 3 and get -\frac{1}{125}.

$$\frac{\frac{2}{25}x^{3}y^{3}}{-\frac{1}{125}x^{3}y^{3}}$$

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

$$\frac{\frac{2}{25}}{-\frac{1}{125}}$$

Divide \frac{2}{25} by -\frac{1}{125} by multiplying \frac{2}{25} by the reciprocal of -\frac{1}{125}.

$$\frac{2}{25}\left(-125\right)$$

Multiply \frac{2}{25} and -125 to get -10.

$$-10$$

Factor

$$-10$$