Solve [1/3a^{2}-5/2a^{2}+1/6a^{2}-a^{2})-(-3/2a^{4}b^{2}+frac{3}{5}a^{4}b^{2}+frac{1}{10}a^{4}b^{2}):(frac{1}{3}a^2b^2+frac{1}{5}a^2b^2)

$$[ \frac { 1 } { 3 } a ^ { 2 } - \frac { 5 } { 2 } a ^ { 2 } + \frac { 1 } { 6 } a ^ { 2 } - a ^ { 2 } ) - ( - \frac { 3 } { 2 } a ^ { 4 } b ^ { 2 } + \frac { 3 } { 5 } a ^ { 4 } b ^ { 2 } + \frac { 1 } { 10 } a ^ { 4 } b ^ { 2 } ) : ( \frac { 1 } { 3 } a ^ { 2 } b ^ { 2 } + \frac { 1 } { 5 } a ^ { 2 } b ^ { 2 } )$$

Evaluate

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

Solution Steps

Combine \frac{1}{3}a^{2} and -\frac{5}{2}a^{2} to get -\frac{13}{6}a^{2}.

$$-\frac{13}{6}a^{2}+\frac{1}{6}a^{2}-a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{13}{6}a^{2} and \frac{1}{6}a^{2} to get -2a^{2}.

$$-2a^{2}-a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

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

$$-3a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{3}{2}a^{4}b^{2} and \frac{3}{5}a^{4}b^{2} to get -\frac{9}{10}a^{4}b^{2}.

$$-3a^{2}-\frac{-\frac{9}{10}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{9}{10}a^{4}b^{2} and \frac{1}{10}a^{4}b^{2} to get -\frac{4}{5}a^{4}b^{2}.

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

Combine \frac{1}{3}a^{2}b^{2} and \frac{1}{5}a^{2}b^{2} to get \frac{8}{15}a^{2}b^{2}.

$$-3a^{2}-\frac{-\frac{4}{5}a^{4}b^{2}}{\frac{8}{15}a^{2}b^{2}}$$

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

$$-3a^{2}-\frac{-\frac{4}{5}a^{2}}{\frac{8}{15}}$$

Divide -\frac{4}{5}a^{2} by \frac{8}{15} by multiplying -\frac{4}{5}a^{2} by the reciprocal of \frac{8}{15}.

$$-3a^{2}-\frac{-\frac{4}{5}a^{2}\times 15}{8}$$

Multiply -\frac{4}{5} and 15 to get -12.

$$-3a^{2}-\frac{-12a^{2}}{8}$$

Divide -12a^{2} by 8 to get -\frac{3}{2}a^{2}.

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

The opposite of -\frac{3}{2}a^{2} is \frac{3}{2}a^{2}.

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

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

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

Expand

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

Solution Steps

Combine \frac{1}{3}a^{2} and -\frac{5}{2}a^{2} to get -\frac{13}{6}a^{2}.

$$-\frac{13}{6}a^{2}+\frac{1}{6}a^{2}-a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{13}{6}a^{2} and \frac{1}{6}a^{2} to get -2a^{2}.

$$-2a^{2}-a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

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

$$-3a^{2}-\frac{-\frac{3}{2}a^{4}b^{2}+\frac{3}{5}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{3}{2}a^{4}b^{2} and \frac{3}{5}a^{4}b^{2} to get -\frac{9}{10}a^{4}b^{2}.

$$-3a^{2}-\frac{-\frac{9}{10}a^{4}b^{2}+\frac{1}{10}a^{4}b^{2}}{\frac{1}{3}a^{2}b^{2}+\frac{1}{5}a^{2}b^{2}}$$

Combine -\frac{9}{10}a^{4}b^{2} and \frac{1}{10}a^{4}b^{2} to get -\frac{4}{5}a^{4}b^{2}.

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

Combine \frac{1}{3}a^{2}b^{2} and \frac{1}{5}a^{2}b^{2} to get \frac{8}{15}a^{2}b^{2}.

$$-3a^{2}-\frac{-\frac{4}{5}a^{4}b^{2}}{\frac{8}{15}a^{2}b^{2}}$$

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

$$-3a^{2}-\frac{-\frac{4}{5}a^{2}}{\frac{8}{15}}$$

Divide -\frac{4}{5}a^{2} by \frac{8}{15} by multiplying -\frac{4}{5}a^{2} by the reciprocal of \frac{8}{15}.

$$-3a^{2}-\frac{-\frac{4}{5}a^{2}\times 15}{8}$$

Multiply -\frac{4}{5} and 15 to get -12.

$$-3a^{2}-\frac{-12a^{2}}{8}$$

Divide -12a^{2} by 8 to get -\frac{3}{2}a^{2}.

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

The opposite of -\frac{3}{2}a^{2} is \frac{3}{2}a^{2}.

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

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

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