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

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

Evaluate

$$\frac{b^{4}a^{10}}{4}$$

Short Solution Steps

Use the distributive property to multiply 2a^{3}b by \frac{1}{4}a^{2}b+\frac{3}{2}ab-b.

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

Use the distributive property to multiply b^{2} by 3a^{4}-2a^{3}.

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

To find the opposite of 3b^{2}a^{4}-2b^{2}a^{3}, find the opposite of each term.

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

Combine 3a^{4}b^{2} and -3b^{2}a^{4} to get 0.

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

Combine -2a^{3}b^{2} and 2b^{2}a^{3} to get 0.

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

Expand \left(\frac{1}{2}a^{5}b^{2}\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{\left(\frac{1}{2}\right)^{3}a^{15}\left(b^{2}\right)^{3}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\left(\frac{1}{2}\right)^{3}a^{15}b^{6}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}a^{15}b^{6}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

Use the distributive property to multiply a^{2}b by \frac{1}{2}a^{3}b+1.

$$\frac{\frac{1}{8}a^{15}b^{6}}{\frac{1}{2}a^{5}b^{2}+a^{2}b-a^{2}b}$$

Combine a^{2}b and -a^{2}b to get 0.

$$\frac{\frac{1}{8}a^{15}b^{6}}{\frac{1}{2}a^{5}b^{2}}$$

Cancel out b^{2}a^{5} in both numerator and denominator.

$$\frac{\frac{1}{8}b^{4}a^{10}}{\frac{1}{2}}$$

Divide \frac{1}{8}b^{4}a^{10} by \frac{1}{2} by multiplying \frac{1}{8}b^{4}a^{10} by the reciprocal of \frac{1}{2}.

$$\frac{1}{8}b^{4}a^{10}\times 2$$

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

$$\frac{1}{4}b^{4}a^{10}$$

Expand

$$\frac{b^{4}a^{10}}{4}$$

Short Solution Steps

Use the distributive property to multiply 2a^{3}b by \frac{1}{4}a^{2}b+\frac{3}{2}ab-b.

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

Use the distributive property to multiply b^{2} by 3a^{4}-2a^{3}.

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

To find the opposite of 3b^{2}a^{4}-2b^{2}a^{3}, find the opposite of each term.

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

Combine 3a^{4}b^{2} and -3b^{2}a^{4} to get 0.

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

Combine -2a^{3}b^{2} and 2b^{2}a^{3} to get 0.

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

Expand \left(\frac{1}{2}a^{5}b^{2}\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{\left(\frac{1}{2}\right)^{3}a^{15}\left(b^{2}\right)^{3}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\left(\frac{1}{2}\right)^{3}a^{15}b^{6}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}a^{15}b^{6}}{a^{2}b\left(\frac{1}{2}a^{3}b+1\right)-a^{2}b}$$

Use the distributive property to multiply a^{2}b by \frac{1}{2}a^{3}b+1.

$$\frac{\frac{1}{8}a^{15}b^{6}}{\frac{1}{2}a^{5}b^{2}+a^{2}b-a^{2}b}$$

Combine a^{2}b and -a^{2}b to get 0.

$$\frac{\frac{1}{8}a^{15}b^{6}}{\frac{1}{2}a^{5}b^{2}}$$

Cancel out b^{2}a^{5} in both numerator and denominator.

$$\frac{\frac{1}{8}b^{4}a^{10}}{\frac{1}{2}}$$

Divide \frac{1}{8}b^{4}a^{10} by \frac{1}{2} by multiplying \frac{1}{8}b^{4}a^{10} by the reciprocal of \frac{1}{2}.

$$\frac{1}{8}b^{4}a^{10}\times 2$$

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

$$\frac{1}{4}b^{4}a^{10}$$