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

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

Evaluate

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

Solution Steps

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

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

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

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

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\frac{x^{7}y^{2}\left(-\frac{1}{4}\right)y^{2}+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}x^{2}y\right)^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}x^{2}y\right)^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}\right)^{2}x^{4}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

Calculate -\frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\times \frac{1}{9}x^{4}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

Multiply \frac{3}{2} and \frac{1}{9} to get \frac{1}{6}.

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

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Combine x^{7}y^{4}\left(-\frac{1}{4}\right) and \frac{1}{6}x^{7}y^{4} to get -\frac{1}{12}x^{7}y^{4}.

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x\left(x^{3}\right)^{2}y^{2}}$$

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

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}xx^{6}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x^{7}y^{2}}$$

Cancel out y^{2}x^{7} in both numerator and denominator.

$$\frac{-\frac{1}{12}y^{2}}{\frac{1}{4}}$$

Divide -\frac{1}{12}y^{2} by \frac{1}{4} by multiplying -\frac{1}{12}y^{2} by the reciprocal of \frac{1}{4}.

$$-\frac{1}{12}y^{2}\times 4$$

Multiply -\frac{1}{12} and 4 to get -\frac{1}{3}.

$$-\frac{1}{3}y^{2}$$

Expand

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

Solution Steps

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

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

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

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

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\frac{x^{7}y^{2}\left(-\frac{1}{4}\right)y^{2}+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}x^{2}y\right)^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}x^{2}y\right)^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\left(-\frac{1}{3}\right)^{2}x^{4}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

Calculate -\frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{x^{7}y^{4}\left(-\frac{1}{4}\right)+\frac{3}{2}x^{3}y^{2}\times \frac{1}{9}x^{4}y^{2}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

Multiply \frac{3}{2} and \frac{1}{9} to get \frac{1}{6}.

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

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Combine x^{7}y^{4}\left(-\frac{1}{4}\right) and \frac{1}{6}x^{7}y^{4} to get -\frac{1}{12}x^{7}y^{4}.

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x\left(x^{3}y\right)^{2}}$$

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

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x\left(x^{3}\right)^{2}y^{2}}$$

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

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}xx^{6}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{-\frac{1}{12}x^{7}y^{4}}{\frac{1}{4}x^{7}y^{2}}$$

Cancel out y^{2}x^{7} in both numerator and denominator.

$$\frac{-\frac{1}{12}y^{2}}{\frac{1}{4}}$$

Divide -\frac{1}{12}y^{2} by \frac{1}{4} by multiplying -\frac{1}{12}y^{2} by the reciprocal of \frac{1}{4}.

$$-\frac{1}{12}y^{2}\times 4$$

Multiply -\frac{1}{12} and 4 to get -\frac{1}{3}.

$$-\frac{1}{3}y^{2}$$