Solve (+1/3)(2a^{2}b+3/2a^{2}b^{2})+(a^{4}b^{4}+4/3a^{4}b^{3}-4a^3b^4):(-2a^2b^2)

$$( + \frac { 1 } { 3 } ) ( 2 a ^ { 2 } b + \frac { 3 } { 2 } a ^ { 2 } b ^ { 2 } ) + ( a ^ { 4 } b ^ { 4 } + \frac { 4 } { 3 } a ^ { 4 } b ^ { 3 } - 4 a ^ { 3 } b ^ { 4 } ) : ( - 2 a ^ { 2 } b ^ { 2 } )$$

Evaluate

$$2ab^{2}$$

Short Solution Steps

Use the distributive property to multiply \frac{1}{3} by 2a^{2}b+\frac{3}{2}a^{2}b^{2}.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{a^{4}b^{4}+\frac{4}{3}a^{4}b^{3}-4a^{3}b^{4}}{-2a^{2}b^{2}}$$

Factor the expressions that are not already factored in \frac{a^{4}b^{4}+\frac{4}{3}a^{4}b^{3}-4a^{3}b^{4}}{-2a^{2}b^{2}}.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{\frac{1}{3}\left(3ab+4a-12b\right)a^{3}b^{3}}{-2a^{2}b^{2}}$$

Cancel out a^{2}b^{2} in both numerator and denominator.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{\frac{1}{3}ab\left(3ab+4a-12b\right)}{-2}$$

Divide \frac{1}{3}ab\left(3ab+4a-12b\right) by -2 to get -\frac{1}{6}ab\left(3ab+4a-12b\right).

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}-\frac{1}{6}ab\left(3ab+4a-12b\right)$$

Use the distributive property to multiply -\frac{1}{6}ab by 3ab+4a-12b.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}-\frac{1}{2}a^{2}b^{2}-\frac{2}{3}a^{2}b+2ab^{2}$$

Combine \frac{1}{2}a^{2}b^{2} and -\frac{1}{2}a^{2}b^{2} to get 0.

$$\frac{2}{3}a^{2}b-\frac{2}{3}a^{2}b+2ab^{2}$$

Combine \frac{2}{3}a^{2}b and -\frac{2}{3}a^{2}b to get 0.

$$2ab^{2}$$

Expand

$$2ab^{2}$$

Short Solution Steps

Use the distributive property to multiply \frac{1}{3} by 2a^{2}b+\frac{3}{2}a^{2}b^{2}.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{a^{4}b^{4}+\frac{4}{3}a^{4}b^{3}-4a^{3}b^{4}}{-2a^{2}b^{2}}$$

Factor the expressions that are not already factored in \frac{a^{4}b^{4}+\frac{4}{3}a^{4}b^{3}-4a^{3}b^{4}}{-2a^{2}b^{2}}.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{\frac{1}{3}\left(3ab+4a-12b\right)a^{3}b^{3}}{-2a^{2}b^{2}}$$

Cancel out a^{2}b^{2} in both numerator and denominator.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}+\frac{\frac{1}{3}ab\left(3ab+4a-12b\right)}{-2}$$

Divide \frac{1}{3}ab\left(3ab+4a-12b\right) by -2 to get -\frac{1}{6}ab\left(3ab+4a-12b\right).

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}-\frac{1}{6}ab\left(3ab+4a-12b\right)$$

Use the distributive property to multiply -\frac{1}{6}ab by 3ab+4a-12b.

$$\frac{2}{3}a^{2}b+\frac{1}{2}a^{2}b^{2}-\frac{1}{2}a^{2}b^{2}-\frac{2}{3}a^{2}b+2ab^{2}$$

Combine \frac{1}{2}a^{2}b^{2} and -\frac{1}{2}a^{2}b^{2} to get 0.

$$\frac{2}{3}a^{2}b-\frac{2}{3}a^{2}b+2ab^{2}$$

Combine \frac{2}{3}a^{2}b and -\frac{2}{3}a^{2}b to get 0.

$$2ab^{2}$$