Solve 1/2[3/5a^{2}b(-1/3ab^{2})-7/5a^{3}(-frac{3}{14}b^{3})+frac{1}{2}a^3b^3]+a^3b^3

$$\frac { 1 } { 2 } [ \frac { 3 } { 5 } a ^ { 2 } b ( - \frac { 1 } { 3 } a b ^ { 2 } ) - \frac { 7 } { 5 } a ^ { 3 } ( - \frac { 3 } { 14 } b ^ { 3 } ) + \frac { 1 } { 2 } a ^ { 3 } b ^ { 3 } ] + a ^ { 3 } b ^ { 3 }$$

Evaluate

$$\frac{13\left(ab\right)^{3}}{10}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}\left(\frac{3}{5}a^{3}b\left(-\frac{1}{3}\right)b^{2}-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{2}\left(\frac{3}{5}a^{3}b^{3}\left(-\frac{1}{3}\right)-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Multiply \frac{3}{5} and -\frac{1}{3} to get -\frac{1}{5}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Multiply \frac{7}{5} and -\frac{3}{14} to get -\frac{3}{10}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}-\left(-\frac{3}{10}a^{3}b^{3}\right)+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

The opposite of -\frac{3}{10}a^{3}b^{3} is \frac{3}{10}a^{3}b^{3}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}+\frac{3}{10}a^{3}b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Combine -\frac{1}{5}a^{3}b^{3} and \frac{3}{10}a^{3}b^{3} to get \frac{1}{10}a^{3}b^{3}.

$$\frac{1}{2}\left(\frac{1}{10}a^{3}b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Combine \frac{1}{10}a^{3}b^{3} and \frac{1}{2}a^{3}b^{3} to get \frac{3}{5}a^{3}b^{3}.

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

Multiply \frac{1}{2} and \frac{3}{5} to get \frac{3}{10}.

$$\frac{3}{10}a^{3}b^{3}+a^{3}b^{3}$$

Combine \frac{3}{10}a^{3}b^{3} and a^{3}b^{3} to get \frac{13}{10}a^{3}b^{3}.

$$\frac{13}{10}a^{3}b^{3}$$

Expand

$$\frac{13\left(ab\right)^{3}}{10}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}\left(\frac{3}{5}a^{3}b\left(-\frac{1}{3}\right)b^{2}-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{2}\left(\frac{3}{5}a^{3}b^{3}\left(-\frac{1}{3}\right)-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Multiply \frac{3}{5} and -\frac{1}{3} to get -\frac{1}{5}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}-\frac{7}{5}a^{3}\left(-\frac{3}{14}\right)b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Multiply \frac{7}{5} and -\frac{3}{14} to get -\frac{3}{10}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}-\left(-\frac{3}{10}a^{3}b^{3}\right)+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

The opposite of -\frac{3}{10}a^{3}b^{3} is \frac{3}{10}a^{3}b^{3}.

$$\frac{1}{2}\left(-\frac{1}{5}a^{3}b^{3}+\frac{3}{10}a^{3}b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Combine -\frac{1}{5}a^{3}b^{3} and \frac{3}{10}a^{3}b^{3} to get \frac{1}{10}a^{3}b^{3}.

$$\frac{1}{2}\left(\frac{1}{10}a^{3}b^{3}+\frac{1}{2}a^{3}b^{3}\right)+a^{3}b^{3}$$

Combine \frac{1}{10}a^{3}b^{3} and \frac{1}{2}a^{3}b^{3} to get \frac{3}{5}a^{3}b^{3}.

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

Multiply \frac{1}{2} and \frac{3}{5} to get \frac{3}{10}.

$$\frac{3}{10}a^{3}b^{3}+a^{3}b^{3}$$

Combine \frac{3}{10}a^{3}b^{3} and a^{3}b^{3} to get \frac{13}{10}a^{3}b^{3}.

$$\frac{13}{10}a^{3}b^{3}$$