Solve [(1/2a^2+1/2a-1)(2a-1/3)-(1/3a^2-frac{3}{4}a+2)(frac{4}{3}a-frac{1}{2})]+frac{13}{24}a

$$[ ( \frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 2 } a - 1 ) ( 2 a - \frac { 1 } { 3 } ) - ( \frac { 1 } { 3 } a ^ { 2 } - \frac { 3 } { 4 } a + 2 ) ( \frac { 4 } { 3 } a - \frac { 1 } { 2 } ) ] + \frac { 13 } { 24 } a$$

Evaluate

$$\frac{5a^{3}}{9}+2a^{2}-\frac{14a}{3}+\frac{4}{3}$$

Solution Steps

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

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\left(\frac{1}{3}a^{2}-\frac{3}{4}a+2\right)\left(\frac{4}{3}a-\frac{1}{2}\right)+\frac{13}{24}a$$

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

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\left(\frac{4}{9}a^{3}-\frac{7}{6}a^{2}+\frac{73}{24}a-1\right)+\frac{13}{24}a$$

To find the opposite of \frac{4}{9}a^{3}-\frac{7}{6}a^{2}+\frac{73}{24}a-1, find the opposite of each term.

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\frac{4}{9}a^{3}+\frac{7}{6}a^{2}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine a^{3} and -\frac{4}{9}a^{3} to get \frac{5}{9}a^{3}.

$$\frac{5}{9}a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}+\frac{7}{6}a^{2}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine \frac{5}{6}a^{2} and \frac{7}{6}a^{2} to get 2a^{2}.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{13}{6}a+\frac{1}{3}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine -\frac{13}{6}a and -\frac{73}{24}a to get -\frac{125}{24}a.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{125}{24}a+\frac{1}{3}+1+\frac{13}{24}a$$

Add \frac{1}{3} and 1 to get \frac{4}{3}.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{125}{24}a+\frac{4}{3}+\frac{13}{24}a$$

Combine -\frac{125}{24}a and \frac{13}{24}a to get -\frac{14}{3}a.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{14}{3}a+\frac{4}{3}$$

Expand

$$\frac{5a^{3}}{9}+2a^{2}-\frac{14a}{3}+\frac{4}{3}$$

Solution Steps

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

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\left(\frac{1}{3}a^{2}-\frac{3}{4}a+2\right)\left(\frac{4}{3}a-\frac{1}{2}\right)+\frac{13}{24}a$$

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

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\left(\frac{4}{9}a^{3}-\frac{7}{6}a^{2}+\frac{73}{24}a-1\right)+\frac{13}{24}a$$

To find the opposite of \frac{4}{9}a^{3}-\frac{7}{6}a^{2}+\frac{73}{24}a-1, find the opposite of each term.

$$a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}-\frac{4}{9}a^{3}+\frac{7}{6}a^{2}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine a^{3} and -\frac{4}{9}a^{3} to get \frac{5}{9}a^{3}.

$$\frac{5}{9}a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a+\frac{1}{3}+\frac{7}{6}a^{2}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine \frac{5}{6}a^{2} and \frac{7}{6}a^{2} to get 2a^{2}.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{13}{6}a+\frac{1}{3}-\frac{73}{24}a+1+\frac{13}{24}a$$

Combine -\frac{13}{6}a and -\frac{73}{24}a to get -\frac{125}{24}a.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{125}{24}a+\frac{1}{3}+1+\frac{13}{24}a$$

Add \frac{1}{3} and 1 to get \frac{4}{3}.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{125}{24}a+\frac{4}{3}+\frac{13}{24}a$$

Combine -\frac{125}{24}a and \frac{13}{24}a to get -\frac{14}{3}a.

$$\frac{5}{9}a^{3}+2a^{2}-\frac{14}{3}a+\frac{4}{3}$$