Solve -(+8/3a^{3}b^{2})(-1/4a^{2}b)^{2}-1/32ab(4a^{2}b)^{3}+a(-a^{3}b)^2(-1/4b^2)+2a^7b^4

$$- ( + \frac { 8 } { 3 } a ^ { 3 } b ^ { 2 } ) ( - \frac { 1 } { 4 } a ^ { 2 } b ) ^ { 2 } - \frac { 1 } { 32 } a b ( 4 a ^ { 2 } b ) ^ { 3 } + a ( - a ^ { 3 } b ) ^ { 2 } ( - \frac { 1 } { 4 } b ^ { 2 } ) + 2 a ^ { 7 } b ^ { 4 }$$

Evaluate

$$-\frac{5b^{4}a^{7}}{12}$$

Solution Steps

Expand \left(-\frac{1}{4}a^{2}b\right)^{2}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\left(-\frac{1}{4}\right)^{2}\left(a^{2}\right)^{2}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\left(-\frac{1}{4}\right)^{2}a^{4}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate -\frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Expand \left(4a^{2}b\right)^{3}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 4^{3}\left(a^{2}\right)^{3}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 4^{3}a^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate 4 to the power of 3 and get 64.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 64a^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Multiply \frac{1}{32} and 64 to get 2.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2aba^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}bb^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(-a^{3}\right)^{2}b^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate -a^{3} to the power of 2 and get \left(a^{3}\right)^{2}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(a^{3}\right)^{2}b^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(a^{3}\right)^{2}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+aa^{6}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a^{7}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

Combine a^{7}b^{4}\left(-\frac{1}{4}\right) and 2a^{7}b^{4} to get \frac{7}{4}a^{7}b^{4}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

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

$$-\frac{8}{3}a^{3}b^{2}\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Multiply -\frac{8}{3} and \frac{1}{16} to get -\frac{1}{6}.

$$-\frac{1}{6}a^{3}b^{2}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$-\frac{1}{6}a^{7}b^{2}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

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

$$-\frac{1}{6}a^{7}b^{4}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Combine -\frac{1}{6}a^{7}b^{4} and -2a^{7}b^{4} to get -\frac{13}{6}a^{7}b^{4}.

$$-\frac{13}{6}a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Combine -\frac{13}{6}a^{7}b^{4} and \frac{7}{4}a^{7}b^{4} to get -\frac{5}{12}a^{7}b^{4}.

$$-\frac{5}{12}a^{7}b^{4}$$

Expand

$$-\frac{5b^{4}a^{7}}{12}$$

Solution Steps

Expand \left(-\frac{1}{4}a^{2}b\right)^{2}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\left(-\frac{1}{4}\right)^{2}\left(a^{2}\right)^{2}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\left(-\frac{1}{4}\right)^{2}a^{4}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate -\frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times \left(4a^{2}b\right)^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Expand \left(4a^{2}b\right)^{3}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 4^{3}\left(a^{2}\right)^{3}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 4^{3}a^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate 4 to the power of 3 and get 64.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-\frac{1}{32}ab\times 64a^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Multiply \frac{1}{32} and 64 to get 2.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2aba^{6}b^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}bb^{3}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(\left(-a^{3}\right)b\right)^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(-a^{3}\right)^{2}b^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

Calculate -a^{3} to the power of 2 and get \left(a^{3}\right)^{2}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(a^{3}\right)^{2}b^{2}\left(-\frac{1}{4}\right)b^{2}+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a\left(a^{3}\right)^{2}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

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

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+aa^{6}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+a^{7}b^{4}\left(-\frac{1}{4}\right)+2a^{7}b^{4}$$

Combine a^{7}b^{4}\left(-\frac{1}{4}\right) and 2a^{7}b^{4} to get \frac{7}{4}a^{7}b^{4}.

$$\left(-\frac{8}{3}a^{3}b^{2}\right)\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

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

$$-\frac{8}{3}a^{3}b^{2}\times \frac{1}{16}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Multiply -\frac{8}{3} and \frac{1}{16} to get -\frac{1}{6}.

$$-\frac{1}{6}a^{3}b^{2}a^{4}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$-\frac{1}{6}a^{7}b^{2}b^{2}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

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

$$-\frac{1}{6}a^{7}b^{4}-2a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Combine -\frac{1}{6}a^{7}b^{4} and -2a^{7}b^{4} to get -\frac{13}{6}a^{7}b^{4}.

$$-\frac{13}{6}a^{7}b^{4}+\frac{7}{4}a^{7}b^{4}$$

Combine -\frac{13}{6}a^{7}b^{4} and \frac{7}{4}a^{7}b^{4} to get -\frac{5}{12}a^{7}b^{4}.

$$-\frac{5}{12}a^{7}b^{4}$$