Solve -2(3/4a^2+2ab)-[-20/3a^2+(2a-1/4ab-1/2b)(3a-2b)-(frac{1}{9}a+b)(3a-frac{3}{2}b)]-frac{3}{2}ab(frac{1}{2}a-frac{1}{3}b)

$$- 2 ( \frac { 3 } { 4 } a ^ { 2 } + 2 a b ) - [ - \frac { 20 } { 3 } a ^ { 2 } + ( 2 a - \frac { 1 } { 4 } a b - \frac { 1 } { 2 } b ) ( 3 a - 2 b ) - ( \frac { 1 } { 9 } a + b ) ( 3 a - \frac { 3 } { 2 } b ) ] - \frac { 3 } { 2 } a b ( \frac { 1 } { 2 } a - \frac { 1 } { 3 } b )$$

Evaluate

$$\frac{13ab}{3}-\frac{5b^{2}}{2}-\frac{a^{2}}{2}$$

Solution Steps

Use the distributive property to multiply -2 by \frac{3}{4}a^{2}+2ab.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{20}{3}a^{2}+\left(2a-\frac{1}{4}ab-\frac{1}{2}b\right)\left(3a-2b\right)-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Use the distributive property to multiply 2a-\frac{1}{4}ab-\frac{1}{2}b by 3a-2b and combine like terms.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{20}{3}a^{2}+6a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -\frac{20}{3}a^{2} and 6a^{2} to get -\frac{2}{3}a^{2}.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Use the distributive property to multiply \frac{1}{9}a+b by 3a-\frac{3}{2}b and combine like terms.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{3}a^{2}+\frac{17}{6}ab-\frac{3}{2}b^{2}\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

To find the opposite of \frac{1}{3}a^{2}+\frac{17}{6}ab-\frac{3}{2}b^{2}, find the opposite of each term.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\frac{1}{3}a^{2}-\frac{17}{6}ab+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\frac{17}{6}ab+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -\frac{11}{2}ab and -\frac{17}{6}ab to get -\frac{25}{3}ab.

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine b^{2} and \frac{3}{2}b^{2} to get \frac{5}{2}b^{2}.

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+\frac{5}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

To find the opposite of -a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+\frac{5}{2}b^{2}, find the opposite of each term.

$$-\frac{3}{2}a^{2}-4ab+a^{2}+\frac{25}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{1}{2}a^{2}-4ab+\frac{25}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -4ab and \frac{25}{3}ab to get \frac{13}{3}ab.

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}$$

Combine \frac{3}{4}a^{2}b and -\frac{3}{4}a^{2}b to get 0.

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}+\frac{1}{2}ab^{2}$$

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

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab-\frac{5}{2}b^{2}$$

Expand

$$\frac{13ab}{3}-\frac{5b^{2}}{2}-\frac{a^{2}}{2}$$

Solution Steps

Use the distributive property to multiply -2 by \frac{3}{4}a^{2}+2ab.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{20}{3}a^{2}+\left(2a-\frac{1}{4}ab-\frac{1}{2}b\right)\left(3a-2b\right)-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Use the distributive property to multiply 2a-\frac{1}{4}ab-\frac{1}{2}b by 3a-2b and combine like terms.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{20}{3}a^{2}+6a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -\frac{20}{3}a^{2} and 6a^{2} to get -\frac{2}{3}a^{2}.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{9}a+b\right)\left(3a-\frac{3}{2}b\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Use the distributive property to multiply \frac{1}{9}a+b by 3a-\frac{3}{2}b and combine like terms.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\left(\frac{1}{3}a^{2}+\frac{17}{6}ab-\frac{3}{2}b^{2}\right)\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

To find the opposite of \frac{1}{3}a^{2}+\frac{17}{6}ab-\frac{3}{2}b^{2}, find the opposite of each term.

$$-\frac{3}{2}a^{2}-4ab-\left(-\frac{2}{3}a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\frac{1}{3}a^{2}-\frac{17}{6}ab+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{11}{2}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}-\frac{17}{6}ab+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -\frac{11}{2}ab and -\frac{17}{6}ab to get -\frac{25}{3}ab.

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+b^{2}+\frac{3}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine b^{2} and \frac{3}{2}b^{2} to get \frac{5}{2}b^{2}.

$$-\frac{3}{2}a^{2}-4ab-\left(-a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+\frac{5}{2}b^{2}\right)-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

To find the opposite of -a^{2}-\frac{25}{3}ab-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}+\frac{5}{2}b^{2}, find the opposite of each term.

$$-\frac{3}{2}a^{2}-4ab+a^{2}+\frac{25}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{1}{2}a^{2}-4ab+\frac{25}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

Combine -4ab and \frac{25}{3}ab to get \frac{13}{3}ab.

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{2}ab\left(\frac{1}{2}a-\frac{1}{3}b\right)$$

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

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab+\frac{3}{4}a^{2}b-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}-\frac{3}{4}a^{2}b+\frac{1}{2}ab^{2}$$

Combine \frac{3}{4}a^{2}b and -\frac{3}{4}a^{2}b to get 0.

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab-\frac{1}{2}ab^{2}-\frac{5}{2}b^{2}+\frac{1}{2}ab^{2}$$

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

$$-\frac{1}{2}a^{2}+\frac{13}{3}ab-\frac{5}{2}b^{2}$$