Solve (6/5abc-3/4abc+3/10abc)*14/3ac^{2}*(a^{2}b^{2}-frac{1}{3}a^2b^2-frac{1}{6}a^2b^2)*(frac{2}{3}a-frac{1}{6}a)

$$( \frac { 6 } { 5 } a b c - \frac { 3 } { 4 } a b c + \frac { 3 } { 10 } a b c ) \cdot \frac { 14 } { 3 } a c ^ { 2 } \cdot ( a ^ { 2 } b ^ { 2 } - \frac { 1 } { 3 } a ^ { 2 } b ^ { 2 } - \frac { 1 } { 6 } a ^ { 2 } b ^ { 2 } ) \cdot ( \frac { 2 } { 3 } a - \frac { 1 } { 6 } a )$$

Evaluate

$$\frac{7a^{5}\left(bc\right)^{3}}{8}$$

Solution Steps

Combine \frac{6}{5}abc and -\frac{3}{4}abc to get \frac{9}{20}abc.

$$\left(\frac{9}{20}abc+\frac{3}{10}abc\right)\times \frac{14}{3}ac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Combine \frac{9}{20}abc and \frac{3}{10}abc to get \frac{3}{4}abc.

$$\frac{3}{4}abc\times \frac{14}{3}ac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply \frac{3}{4} and \frac{14}{3} to get \frac{7}{2}.

$$\frac{7}{2}abcac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply a and a to get a^{2}.

$$\frac{7}{2}a^{2}bcc^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\left(\frac{2}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\times \frac{1}{2}a^{2}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply \frac{7}{2} and \frac{1}{2} to get \frac{7}{4}.

$$\frac{7}{4}a^{2}bc^{3}a^{2}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{7}{4}a^{4}bc^{3}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{4}a^{4}b^{3}c^{3}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{4}a^{4}b^{3}c^{3}\times \frac{1}{2}a$$

Multiply \frac{7}{4} and \frac{1}{2} to get \frac{7}{8}.

$$\frac{7}{8}a^{4}b^{3}c^{3}a$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{7}{8}a^{5}b^{3}c^{3}$$

Expand

$$\frac{7a^{5}\left(bc\right)^{3}}{8}$$

Solution Steps

Combine \frac{6}{5}abc and -\frac{3}{4}abc to get \frac{9}{20}abc.

$$\left(\frac{9}{20}abc+\frac{3}{10}abc\right)\times \frac{14}{3}ac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Combine \frac{9}{20}abc and \frac{3}{10}abc to get \frac{3}{4}abc.

$$\frac{3}{4}abc\times \frac{14}{3}ac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply \frac{3}{4} and \frac{14}{3} to get \frac{7}{2}.

$$\frac{7}{2}abcac^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply a and a to get a^{2}.

$$\frac{7}{2}a^{2}bcc^{2}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\left(a^{2}b^{2}-\frac{1}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\left(\frac{2}{3}a^{2}b^{2}-\frac{1}{6}a^{2}b^{2}\right)\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{2}a^{2}bc^{3}\times \frac{1}{2}a^{2}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

Multiply \frac{7}{2} and \frac{1}{2} to get \frac{7}{4}.

$$\frac{7}{4}a^{2}bc^{3}a^{2}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{7}{4}a^{4}bc^{3}b^{2}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{4}a^{4}b^{3}c^{3}\left(\frac{2}{3}a-\frac{1}{6}a\right)$$

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

$$\frac{7}{4}a^{4}b^{3}c^{3}\times \frac{1}{2}a$$

Multiply \frac{7}{4} and \frac{1}{2} to get \frac{7}{8}.

$$\frac{7}{8}a^{4}b^{3}c^{3}a$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{7}{8}a^{5}b^{3}c^{3}$$