Solve 11(-2/3a^{2})^{2}:(-2/3a)^3+4/3a^3b^2:(-1/3ab)^2

$$11 ( - \frac { 2 } { 3 } a ^ { 2 } ) ^ { 2 } : ( - \frac { 2 } { 3 } a ) ^ { 3 } + \frac { 4 } { 3 } a ^ { 3 } b ^ { 2 } : ( - \frac { 1 } { 3 } a b ) ^ { 2 }$$

Evaluate

$$-\frac{9a}{2}$$

Solution Steps

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

$$\frac{11\left(-\frac{2}{3}\right)^{2}\left(a^{2}\right)^{2}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$\frac{11\left(-\frac{2}{3}\right)^{2}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{11\times \frac{4}{9}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Multiply 11 and \frac{4}{9} to get \frac{44}{9}.

$$\frac{\frac{44}{9}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$\frac{\frac{44}{9}a^{4}}{\left(-\frac{2}{3}\right)^{3}a^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 3 and get -\frac{8}{27}.

$$\frac{\frac{44}{9}a^{4}}{-\frac{8}{27}a^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Cancel out a^{3} in both numerator and denominator.

$$\frac{\frac{44}{9}a}{-\frac{8}{27}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Divide \frac{44}{9}a by -\frac{8}{27} by multiplying \frac{44}{9}a by the reciprocal of -\frac{8}{27}.

$$\frac{\frac{44}{9}a\times 27}{-8}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Multiply \frac{44}{9} and 27 to get 132.

$$\frac{132a}{-8}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Divide 132a by -8 to get -\frac{33}{2}a.

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}\right)^{2}a^{2}b^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\frac{1}{9}a^{2}b^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a}{\frac{1}{9}}$$

Divide \frac{4}{3}a by \frac{1}{9} by multiplying \frac{4}{3}a by the reciprocal of \frac{1}{9}.

$$-\frac{33}{2}a+\frac{4}{3}a\times 9$$

Multiply \frac{4}{3} and 9 to get 12.

$$-\frac{33}{2}a+12a$$

Combine -\frac{33}{2}a and 12a to get -\frac{9}{2}a.

$$-\frac{9}{2}a$$

Expand

$$-\frac{9a}{2}$$

Solution Steps

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

$$\frac{11\left(-\frac{2}{3}\right)^{2}\left(a^{2}\right)^{2}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$\frac{11\left(-\frac{2}{3}\right)^{2}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{11\times \frac{4}{9}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Multiply 11 and \frac{4}{9} to get \frac{44}{9}.

$$\frac{\frac{44}{9}a^{4}}{\left(-\frac{2}{3}a\right)^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$\frac{\frac{44}{9}a^{4}}{\left(-\frac{2}{3}\right)^{3}a^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Calculate -\frac{2}{3} to the power of 3 and get -\frac{8}{27}.

$$\frac{\frac{44}{9}a^{4}}{-\frac{8}{27}a^{3}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Cancel out a^{3} in both numerator and denominator.

$$\frac{\frac{44}{9}a}{-\frac{8}{27}}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Divide \frac{44}{9}a by -\frac{8}{27} by multiplying \frac{44}{9}a by the reciprocal of -\frac{8}{27}.

$$\frac{\frac{44}{9}a\times 27}{-8}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Multiply \frac{44}{9} and 27 to get 132.

$$\frac{132a}{-8}+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

Divide 132a by -8 to get -\frac{33}{2}a.

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}ab\right)^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\left(-\frac{1}{3}\right)^{2}a^{2}b^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a^{3}b^{2}}{\frac{1}{9}a^{2}b^{2}}$$

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

$$-\frac{33}{2}a+\frac{\frac{4}{3}a}{\frac{1}{9}}$$

Divide \frac{4}{3}a by \frac{1}{9} by multiplying \frac{4}{3}a by the reciprocal of \frac{1}{9}.

$$-\frac{33}{2}a+\frac{4}{3}a\times 9$$

Multiply \frac{4}{3} and 9 to get 12.

$$-\frac{33}{2}a+12a$$

Combine -\frac{33}{2}a and 12a to get -\frac{9}{2}a.

$$-\frac{9}{2}a$$