Solve [(3x^{2}y-5/2x^{2}y)^{3}:(-1/4x^{4}y^2)-(+1/3y)(-x^2)](-xy^2)+(1/2xy)[frac{5}{6}xy+frac{1}{6}xy]^2

$$[ ( 3 x ^ { 2 } y - \frac { 5 } { 2 } x ^ { 2 } y ) ^ { 3 } : ( - \frac { 1 } { 4 } x ^ { 4 } y ^ { 2 } ) - ( + \frac { 1 } { 3 } y ) ( - x ^ { 2 } ) ] ( - x y ^ { 2 } ) + ( \frac { 1 } { 2 } x y ) [ \frac { 5 } { 6 } x y + \frac { 1 } { 6 } x y ] ^ { 2 }$$

Evaluate

$$\frac{2\left(xy\right)^{3}}{3}$$

Solution Steps

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

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

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

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

Cancel out y^{2}x^{4} in both numerator and denominator.

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

Divide \frac{1}{8}yx^{2} by -\frac{1}{4} by multiplying \frac{1}{8}yx^{2} by the reciprocal of -\frac{1}{4}.

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

Multiply \frac{1}{8} and 4 to get \frac{1}{2}.

$$\left(\frac{\frac{1}{2}yx^{2}}{-1}-\frac{1}{3}y\left(-x^{2}\right)\right)\left(-x\right)y^{2}+\frac{1}{2}xy\left(\frac{5}{6}xy+\frac{1}{6}xy\right)^{2}$$

Anything divided by -1 gives its opposite.

$$\left(-\frac{1}{2}yx^{2}-\frac{1}{3}y\left(-x^{2}\right)\right)\left(-x\right)y^{2}+\frac{1}{2}xy\left(\frac{5}{6}xy+\frac{1}{6}xy\right)^{2}$$

Combine \frac{5}{6}xy and \frac{1}{6}xy to get xy.

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

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

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

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

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

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

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

Multiply -1 and \frac{1}{3} to get -\frac{1}{3}.

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

Multiply -\frac{1}{3} and -1 to get \frac{1}{3}.

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

Combine -\frac{1}{2}yx^{2} and \frac{1}{3}yx^{2} to get -\frac{1}{6}yx^{2}.

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

Multiply -\frac{1}{6} and -1 to get \frac{1}{6}.

$$\frac{1}{6}yx^{2}xy^{2}+\frac{1}{2}x^{3}y^{3}$$

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

$$\frac{1}{6}yx^{3}y^{2}+\frac{1}{2}x^{3}y^{3}$$

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

$$\frac{1}{6}y^{3}x^{3}+\frac{1}{2}x^{3}y^{3}$$

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

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

Expand

$$\frac{2\left(xy\right)^{3}}{3}$$

Solution Steps

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

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

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

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

Cancel out y^{2}x^{4} in both numerator and denominator.

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

Divide \frac{1}{8}yx^{2} by -\frac{1}{4} by multiplying \frac{1}{8}yx^{2} by the reciprocal of -\frac{1}{4}.

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

Multiply \frac{1}{8} and 4 to get \frac{1}{2}.

$$\left(\frac{\frac{1}{2}yx^{2}}{-1}-\frac{1}{3}y\left(-x^{2}\right)\right)\left(-x\right)y^{2}+\frac{1}{2}xy\left(\frac{5}{6}xy+\frac{1}{6}xy\right)^{2}$$

Anything divided by -1 gives its opposite.

$$\left(-\frac{1}{2}yx^{2}-\frac{1}{3}y\left(-x^{2}\right)\right)\left(-x\right)y^{2}+\frac{1}{2}xy\left(\frac{5}{6}xy+\frac{1}{6}xy\right)^{2}$$

Combine \frac{5}{6}xy and \frac{1}{6}xy to get xy.

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

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

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

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

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

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

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

Multiply -1 and \frac{1}{3} to get -\frac{1}{3}.

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

Multiply -\frac{1}{3} and -1 to get \frac{1}{3}.

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

Combine -\frac{1}{2}yx^{2} and \frac{1}{3}yx^{2} to get -\frac{1}{6}yx^{2}.

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

Multiply -\frac{1}{6} and -1 to get \frac{1}{6}.

$$\frac{1}{6}yx^{2}xy^{2}+\frac{1}{2}x^{3}y^{3}$$

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

$$\frac{1}{6}yx^{3}y^{2}+\frac{1}{2}x^{3}y^{3}$$

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

$$\frac{1}{6}y^{3}x^{3}+\frac{1}{2}x^{3}y^{3}$$

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

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