Solve ((2/3x^{2}y^{2}-3/2x^{2}y^{2})^{2}div(1/4xy-7/8xy)^{2}-(frac{5}{3}x^2y^2-frac{1}{6}x^2y^2))*(frac{4}{3}xy-frac{2}{5}xy)

$$( ( \frac { 2 } { 3 } x ^ { 2 } y ^ { 2 } - \frac { 3 } { 2 } x ^ { 2 } y ^ { 2 } ) ^ { 2 } \div ( \frac { 1 } { 4 } x y - \frac { 7 } { 8 } x y ) ^ { 2 } - ( \frac { 5 } { 3 } x ^ { 2 } y ^ { 2 } - \frac { 1 } { 6 } x ^ { 2 } y ^ { 2 } ) ) \times ( \frac { 4 } { 3 } x y - \frac { 2 } { 5 } x y )$$

Evaluate

$$\frac{7\left(xy\right)^{3}}{27}$$

Solution Steps

Combine \frac{2}{3}x^{2}y^{2} and -\frac{3}{2}x^{2}y^{2} to get -\frac{5}{6}x^{2}y^{2}.

$$\left(\frac{\left(-\frac{5}{6}x^{2}y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Expand \left(-\frac{5}{6}x^{2}y^{2}\right)^{2}.

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}\left(x^{2}\right)^{2}\left(y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}x^{4}\left(y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}x^{4}y^{4}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Calculate -\frac{5}{6} to the power of 2 and get \frac{25}{36}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{1}{4}xy and -\frac{7}{8}xy to get -\frac{5}{8}xy.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(-\frac{5}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Expand \left(-\frac{5}{8}xy\right)^{2}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(-\frac{5}{8}\right)^{2}x^{2}y^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Calculate -\frac{5}{8} to the power of 2 and get \frac{25}{64}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\frac{25}{64}x^{2}y^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\frac{25}{36}x^{2}y^{2}}{\frac{25}{64}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Divide \frac{25}{36}x^{2}y^{2} by \frac{25}{64} by multiplying \frac{25}{36}x^{2}y^{2} by the reciprocal of \frac{25}{64}.

$$\left(\frac{\frac{25}{36}x^{2}y^{2}\times 64}{25}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Multiply \frac{25}{36} and 64 to get \frac{400}{9}.

$$\left(\frac{\frac{400}{9}x^{2}y^{2}}{25}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Divide \frac{400}{9}x^{2}y^{2} by 25 to get \frac{16}{9}x^{2}y^{2}.

$$\left(\frac{16}{9}x^{2}y^{2}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{5}{3}x^{2}y^{2} and -\frac{1}{6}x^{2}y^{2} to get \frac{3}{2}x^{2}y^{2}.

$$\left(\frac{16}{9}x^{2}y^{2}-\frac{3}{2}x^{2}y^{2}\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{16}{9}x^{2}y^{2} and -\frac{3}{2}x^{2}y^{2} to get \frac{5}{18}x^{2}y^{2}.

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

Combine \frac{4}{3}xy and -\frac{2}{5}xy to get \frac{14}{15}xy.

$$\frac{5}{18}x^{2}y^{2}\times \frac{14}{15}xy$$

Multiply \frac{5}{18} and \frac{14}{15} to get \frac{7}{27}.

$$\frac{7}{27}x^{2}y^{2}xy$$

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

$$\frac{7}{27}x^{3}y^{2}y$$

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

$$\frac{7}{27}x^{3}y^{3}$$

Expand

$$\frac{7\left(xy\right)^{3}}{27}$$

Solution Steps

Combine \frac{2}{3}x^{2}y^{2} and -\frac{3}{2}x^{2}y^{2} to get -\frac{5}{6}x^{2}y^{2}.

$$\left(\frac{\left(-\frac{5}{6}x^{2}y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Expand \left(-\frac{5}{6}x^{2}y^{2}\right)^{2}.

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}\left(x^{2}\right)^{2}\left(y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}x^{4}\left(y^{2}\right)^{2}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\left(-\frac{5}{6}\right)^{2}x^{4}y^{4}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Calculate -\frac{5}{6} to the power of 2 and get \frac{25}{36}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(\frac{1}{4}xy-\frac{7}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{1}{4}xy and -\frac{7}{8}xy to get -\frac{5}{8}xy.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(-\frac{5}{8}xy\right)^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Expand \left(-\frac{5}{8}xy\right)^{2}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\left(-\frac{5}{8}\right)^{2}x^{2}y^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Calculate -\frac{5}{8} to the power of 2 and get \frac{25}{64}.

$$\left(\frac{\frac{25}{36}x^{4}y^{4}}{\frac{25}{64}x^{2}y^{2}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

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

$$\left(\frac{\frac{25}{36}x^{2}y^{2}}{\frac{25}{64}}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Divide \frac{25}{36}x^{2}y^{2} by \frac{25}{64} by multiplying \frac{25}{36}x^{2}y^{2} by the reciprocal of \frac{25}{64}.

$$\left(\frac{\frac{25}{36}x^{2}y^{2}\times 64}{25}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Multiply \frac{25}{36} and 64 to get \frac{400}{9}.

$$\left(\frac{\frac{400}{9}x^{2}y^{2}}{25}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Divide \frac{400}{9}x^{2}y^{2} by 25 to get \frac{16}{9}x^{2}y^{2}.

$$\left(\frac{16}{9}x^{2}y^{2}-\left(\frac{5}{3}x^{2}y^{2}-\frac{1}{6}x^{2}y^{2}\right)\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{5}{3}x^{2}y^{2} and -\frac{1}{6}x^{2}y^{2} to get \frac{3}{2}x^{2}y^{2}.

$$\left(\frac{16}{9}x^{2}y^{2}-\frac{3}{2}x^{2}y^{2}\right)\left(\frac{4}{3}xy-\frac{2}{5}xy\right)$$

Combine \frac{16}{9}x^{2}y^{2} and -\frac{3}{2}x^{2}y^{2} to get \frac{5}{18}x^{2}y^{2}.

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

Combine \frac{4}{3}xy and -\frac{2}{5}xy to get \frac{14}{15}xy.

$$\frac{5}{18}x^{2}y^{2}\times \frac{14}{15}xy$$

Multiply \frac{5}{18} and \frac{14}{15} to get \frac{7}{27}.

$$\frac{7}{27}x^{2}y^{2}xy$$

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

$$\frac{7}{27}x^{3}y^{2}y$$

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

$$\frac{7}{27}x^{3}y^{3}$$