Solve -(-3ab)(1/9a^{2}b^{3})-9/5a^{2}b(-15/2ab^{3})+8/3ab^3(-frac{1}{4}a^2b)-frac{23}{2}a(a^2b^4)

$$- ( - 3 a b ) ( \frac { 1 } { 9 } a ^ { 2 } b ^ { 3 } ) - \frac { 9 } { 5 } a ^ { 2 } b ( - \frac { 15 } { 2 } a b ^ { 3 } ) + \frac { 8 } { 3 } a b ^ { 3 } ( - \frac { 1 } { 4 } a ^ { 2 } b ) - \frac { 23 } { 2 } a ( a ^ { 2 } b ^ { 4 } )$$

Evaluate

$$\frac{5a^{3}b^{4}}{3}$$

Solution Steps

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

$$\left(-\left(-3ab\right)\right)\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b\left(-\frac{15}{2}\right)b^{3}+\frac{8}{3}ab^{3}\left(-\frac{1}{4}\right)a^{2}b-\frac{23}{2}aa^{2}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\left(-\left(-3ab\right)\right)\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}ab^{3}\left(-\frac{1}{4}\right)a^{2}b-\frac{23}{2}aa^{2}b^{4}$$

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

$$\left(-\left(-3ab\right)\right)\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{3}\left(-\frac{1}{4}\right)b-\frac{23}{2}aa^{2}b^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\left(-\left(-3ab\right)\right)\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}aa^{2}b^{4}$$

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

$$\left(-\left(-3ab\right)\right)\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

The opposite of -3ab is 3ab.

$$3ab\times \frac{1}{9}a^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

Multiply 3 and \frac{1}{9} to get \frac{1}{3}.

$$\frac{1}{3}aba^{2}b^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

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

$$\frac{1}{3}a^{3}bb^{3}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{1}{3}a^{3}b^{4}-\frac{9}{5}a^{3}b^{4}\left(-\frac{15}{2}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

Multiply \frac{9}{5} and -\frac{15}{2} to get -\frac{27}{2}.

$$\frac{1}{3}a^{3}b^{4}-\left(-\frac{27}{2}a^{3}b^{4}\right)+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

The opposite of -\frac{27}{2}a^{3}b^{4} is \frac{27}{2}a^{3}b^{4}.

$$\frac{1}{3}a^{3}b^{4}+\frac{27}{2}a^{3}b^{4}+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

Combine \frac{1}{3}a^{3}b^{4} and \frac{27}{2}a^{3}b^{4} to get \frac{83}{6}a^{3}b^{4}.

$$\frac{83}{6}a^{3}b^{4}+\frac{8}{3}a^{3}b^{4}\left(-\frac{1}{4}\right)-\frac{23}{2}a^{3}b^{4}$$

Multiply \frac{8}{3} and -\frac{1}{4} to get -\frac{2}{3}.

$$\frac{83}{6}a^{3}b^{4}-\frac{2}{3}a^{3}b^{4}-\frac{23}{2}a^{3}b^{4}$$

Combine \frac{83}{6}a^{3}b^{4} and -\frac{2}{3}a^{3}b^{4} to get \frac{79}{6}a^{3}b^{4}.

$$\frac{79}{6}a^{3}b^{4}-\frac{23}{2}a^{3}b^{4}$$

Combine \frac{79}{6}a^{3}b^{4} and -\frac{23}{2}a^{3}b^{4} to get \frac{5}{3}a^{3}b^{4}.

$$\frac{5}{3}a^{3}b^{4}$$

Differentiate w.r.t. a

$$5a^{2}b^{4}$$