Solve [(-a^{3}b+8/5a^{3}b)(-10/9ab^{3}):(+4/9a^2b^2)+(-1/3ab)^2]^2

$$[ ( - a ^ { 3 } b + \frac { 8 } { 5 } a ^ { 3 } b ) ( - \frac { 10 } { 9 } a b ^ { 3 } ) : ( + \frac { 4 } { 9 } a ^ { 2 } b ^ { 2 } ) + ( - \frac { 1 } { 3 } a b ) ^ { 2 } ] ^ { 2 }$$

Evaluate

$$\frac{625\left(ab\right)^{4}}{324}$$

Short Solution Steps

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

$$\left(\frac{-\frac{10}{9}b\left(b\left(-a^{3}\right)+\frac{8}{5}ba^{3}\right)}{\frac{4}{9}a}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Factor the expressions that are not already factored in \frac{-\frac{10}{9}b\left(b\left(-a^{3}\right)+\frac{8}{5}ba^{3}\right)}{\frac{4}{9}a}.

$$\left(\frac{-\frac{10}{9}\times \frac{1}{5}\left(8-5\right)b^{2}a^{3}}{\frac{4}{9}a}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Cancel out a in both numerator and denominator.

$$\left(\frac{-\frac{10}{9}\times \frac{1}{5}\left(8-5\right)a^{2}b^{2}}{\frac{4}{9}}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Multiply -\frac{10}{9} and \frac{1}{5} to get -\frac{2}{9}.

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

Subtract 5 from 8 to get 3.

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

Multiply -\frac{2}{9} and 3 to get -\frac{2}{3}.

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

Divide -\frac{2}{3}a^{2}b^{2} by \frac{4}{9} by multiplying -\frac{2}{3}a^{2}b^{2} by the reciprocal of \frac{4}{9}.

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

Divide -\frac{2}{3}a^{2}b^{2}\times 9 by 4 to get -\frac{1}{6}a^{2}b^{2}\times 9.

$$\left(-\frac{1}{6}a^{2}b^{2}\times 9+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

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

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

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

$$\left(-\frac{1}{6}a^{2}b^{2}\times 9+\frac{1}{9}a^{2}b^{2}\right)^{2}$$

Multiply -\frac{1}{6} and 9 to get -\frac{3}{2}.

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

Combine -\frac{3}{2}a^{2}b^{2} and \frac{1}{9}a^{2}b^{2} to get -\frac{25}{18}a^{2}b^{2}.

$$\left(-\frac{25}{18}a^{2}b^{2}\right)^{2}$$

Expand \left(-\frac{25}{18}a^{2}b^{2}\right)^{2}.

$$\left(-\frac{25}{18}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-\frac{25}{18}\right)^{2}a^{4}\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-\frac{25}{18}\right)^{2}a^{4}b^{4}$$

Calculate -\frac{25}{18} to the power of 2 and get \frac{625}{324}.

$$\frac{625}{324}a^{4}b^{4}$$

Expand

$$\frac{625\left(ab\right)^{4}}{324}$$

Short Solution Steps

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

$$\left(\frac{-\frac{10}{9}b\left(b\left(-a^{3}\right)+\frac{8}{5}ba^{3}\right)}{\frac{4}{9}a}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Factor the expressions that are not already factored in \frac{-\frac{10}{9}b\left(b\left(-a^{3}\right)+\frac{8}{5}ba^{3}\right)}{\frac{4}{9}a}.

$$\left(\frac{-\frac{10}{9}\times \frac{1}{5}\left(8-5\right)b^{2}a^{3}}{\frac{4}{9}a}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Cancel out a in both numerator and denominator.

$$\left(\frac{-\frac{10}{9}\times \frac{1}{5}\left(8-5\right)a^{2}b^{2}}{\frac{4}{9}}+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

Multiply -\frac{10}{9} and \frac{1}{5} to get -\frac{2}{9}.

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

Subtract 5 from 8 to get 3.

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

Multiply -\frac{2}{9} and 3 to get -\frac{2}{3}.

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

Divide -\frac{2}{3}a^{2}b^{2} by \frac{4}{9} by multiplying -\frac{2}{3}a^{2}b^{2} by the reciprocal of \frac{4}{9}.

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

Divide -\frac{2}{3}a^{2}b^{2}\times 9 by 4 to get -\frac{1}{6}a^{2}b^{2}\times 9.

$$\left(-\frac{1}{6}a^{2}b^{2}\times 9+\left(-\frac{1}{3}ab\right)^{2}\right)^{2}$$

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

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

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

$$\left(-\frac{1}{6}a^{2}b^{2}\times 9+\frac{1}{9}a^{2}b^{2}\right)^{2}$$

Multiply -\frac{1}{6} and 9 to get -\frac{3}{2}.

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

Combine -\frac{3}{2}a^{2}b^{2} and \frac{1}{9}a^{2}b^{2} to get -\frac{25}{18}a^{2}b^{2}.

$$\left(-\frac{25}{18}a^{2}b^{2}\right)^{2}$$

Expand \left(-\frac{25}{18}a^{2}b^{2}\right)^{2}.

$$\left(-\frac{25}{18}\right)^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-\frac{25}{18}\right)^{2}a^{4}\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-\frac{25}{18}\right)^{2}a^{4}b^{4}$$

Calculate -\frac{25}{18} to the power of 2 and get \frac{625}{324}.

$$\frac{625}{324}a^{4}b^{4}$$