Solve [(-1/2a^{3}b)^{3}:(1/4a^{6}b^{2})+(-2ab)^{3}:(-2b^2)][(-3ab)(-2b^3)-(-2ab^2)(-2b^2)]

$$[ ( - \frac { 1 } { 2 } a ^ { 3 } b ) ^ { 3 } : ( \frac { 1 } { 4 } a ^ { 6 } b ^ { 2 } ) + ( - 2 a b ) ^ { 3 } : ( - 2 b ^ { 2 } ) ] [ ( - 3 a b ) ( - 2 b ^ { 3 } ) - ( - 2 a b ^ { 2 } ) ( - 2 b ^ { 2 } ) ]$$

Evaluate

$$7a^{4}b^{5}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\left(\frac{\left(-\frac{1}{2}\right)^{3}a^{9}b^{3}}{\frac{1}{4}a^{6}b^{2}}+\frac{\left(-2ab\right)^{3}}{-2b^{2}}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

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

$$\left(\frac{-\frac{1}{8}a^{9}b^{3}}{\frac{1}{4}a^{6}b^{2}}+\frac{\left(-2ab\right)^{3}}{-2b^{2}}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Cancel out b^{2}a^{6} in both numerator and denominator.

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

Divide -\frac{1}{8}ba^{3} by \frac{1}{4} by multiplying -\frac{1}{8}ba^{3} by the reciprocal of \frac{1}{4}.

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

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

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

Calculate -2 to the power of 3 and get -8.

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

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

$$\left(-\frac{1}{8}ba^{3}\times 4+\frac{-4ba^{3}}{-1}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Anything divided by -1 gives its opposite.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Multiply -3 and -2 to get 6.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(6ab^{4}-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Multiply -2 and -2 to get 4.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(6ab^{4}-4ab^{4}\right)$$

Combine 6ab^{4} and -4ab^{4} to get 2ab^{4}.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\times 2ab^{4}$$

Use the distributive property to multiply -\frac{1}{8}ba^{3}\times 4+4ba^{3} by 2.

$$7ba^{3}ab^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$7b^{5}a^{3}a$$

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

$$7b^{5}a^{4}$$

Expand

$$7a^{4}b^{5}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\left(\frac{\left(-\frac{1}{2}\right)^{3}a^{9}b^{3}}{\frac{1}{4}a^{6}b^{2}}+\frac{\left(-2ab\right)^{3}}{-2b^{2}}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

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

$$\left(\frac{-\frac{1}{8}a^{9}b^{3}}{\frac{1}{4}a^{6}b^{2}}+\frac{\left(-2ab\right)^{3}}{-2b^{2}}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Cancel out b^{2}a^{6} in both numerator and denominator.

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

Divide -\frac{1}{8}ba^{3} by \frac{1}{4} by multiplying -\frac{1}{8}ba^{3} by the reciprocal of \frac{1}{4}.

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

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

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

Calculate -2 to the power of 3 and get -8.

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

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

$$\left(-\frac{1}{8}ba^{3}\times 4+\frac{-4ba^{3}}{-1}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Anything divided by -1 gives its opposite.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(-3ab^{4}\left(-2\right)-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Multiply -3 and -2 to get 6.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(6ab^{4}-\left(-2ab^{4}\left(-2\right)\right)\right)$$

Multiply -2 and -2 to get 4.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\left(6ab^{4}-4ab^{4}\right)$$

Combine 6ab^{4} and -4ab^{4} to get 2ab^{4}.

$$\left(-\frac{1}{8}ba^{3}\times 4+4ba^{3}\right)\times 2ab^{4}$$

Use the distributive property to multiply -\frac{1}{8}ba^{3}\times 4+4ba^{3} by 2.

$$7ba^{3}ab^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$7b^{5}a^{3}a$$

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

$$7b^{5}a^{4}$$