Solve (-5/3ab^{2})(5/6a^2b-1/3a^2b+1/2a^2b)+(-frac{1}{3}ab)^2(frac{3}{2}ab-frac{3}{5}ab)]:(frac{47}{5}a)

$$( - \frac { 5 } { 3 } a b ^ { 2 } ) ( \frac { 5 } { 6 } a ^ { 2 } b - \frac { 1 } { 3 } a ^ { 2 } b + \frac { 1 } { 2 } a ^ { 2 } b ) + ( - \frac { 1 } { 3 } a b ) ^ { 2 } ( \frac { 3 } { 2 } a b - \frac { 3 } { 5 } a b ) ] : ( \frac { 47 } { 5 } a )$$

Evaluate

$$-\frac{47\left(ab\right)^{3}}{30}$$

Solution Steps

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

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

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

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

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

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

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

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

Expand \left(-\frac{1}{3}ab\right)^{2}.

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

Calculate -\frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$-\frac{5}{3}a^{3}b^{3}+\frac{1}{9}a^{2}b^{2}\left(\frac{3}{2}ab-\frac{3}{5}ab\right)$$

Combine \frac{3}{2}ab and -\frac{3}{5}ab to get \frac{9}{10}ab.

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

Multiply \frac{1}{9} and \frac{9}{10} to get \frac{1}{10}.

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

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

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

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

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

Combine -\frac{5}{3}a^{3}b^{3} and \frac{1}{10}a^{3}b^{3} to get -\frac{47}{30}a^{3}b^{3}.

$$-\frac{47}{30}a^{3}b^{3}$$

Expand

$$-\frac{47\left(ab\right)^{3}}{30}$$

Solution Steps

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

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

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

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

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

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

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

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

Expand \left(-\frac{1}{3}ab\right)^{2}.

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

Calculate -\frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$-\frac{5}{3}a^{3}b^{3}+\frac{1}{9}a^{2}b^{2}\left(\frac{3}{2}ab-\frac{3}{5}ab\right)$$

Combine \frac{3}{2}ab and -\frac{3}{5}ab to get \frac{9}{10}ab.

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

Multiply \frac{1}{9} and \frac{9}{10} to get \frac{1}{10}.

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

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

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

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

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

Combine -\frac{5}{3}a^{3}b^{3} and \frac{1}{10}a^{3}b^{3} to get -\frac{47}{30}a^{3}b^{3}.

$$-\frac{47}{30}a^{3}b^{3}$$