Solve 1/2a^{4}b^{4}-(1/2a^{3}b-1/4b^{2})(2ab^{3}+2a^{4}b^2)-2a^5b(-1/2a^2b^2)

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

Evaluate

$$\frac{ab^{5}}{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

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

$$\frac{1}{2}a^{4}b^{4}-\left(\frac{1}{2}a^{4}b^{4}+a^{7}b^{3}-\frac{1}{2}b^{5}a\right)-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

To find the opposite of \frac{1}{2}a^{4}b^{4}+a^{7}b^{3}-\frac{1}{2}b^{5}a, find the opposite of each term.

$$\frac{1}{2}a^{4}b^{4}-\frac{1}{2}a^{4}b^{4}-a^{7}b^{3}+\frac{1}{2}b^{5}a-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}a^{4}b^{4} and -\frac{1}{2}a^{4}b^{4} to get 0.

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

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

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a-\left(-a^{7}b^{3}\right)$$

The opposite of -a^{7}b^{3} is a^{7}b^{3}.

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a+a^{7}b^{3}$$

Combine -a^{7}b^{3} and a^{7}b^{3} to get 0.

$$\frac{1}{2}b^{5}a$$

Expand

$$\frac{ab^{5}}{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

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

$$\frac{1}{2}a^{4}b^{4}-\left(\frac{1}{2}a^{4}b^{4}+a^{7}b^{3}-\frac{1}{2}b^{5}a\right)-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

To find the opposite of \frac{1}{2}a^{4}b^{4}+a^{7}b^{3}-\frac{1}{2}b^{5}a, find the opposite of each term.

$$\frac{1}{2}a^{4}b^{4}-\frac{1}{2}a^{4}b^{4}-a^{7}b^{3}+\frac{1}{2}b^{5}a-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}a^{4}b^{4} and -\frac{1}{2}a^{4}b^{4} to get 0.

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a-2a^{7}b^{3}\left(-\frac{1}{2}\right)$$

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

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a-\left(-a^{7}b^{3}\right)$$

The opposite of -a^{7}b^{3} is a^{7}b^{3}.

$$-a^{7}b^{3}+\frac{1}{2}b^{5}a+a^{7}b^{3}$$

Combine -a^{7}b^{3} and a^{7}b^{3} to get 0.

$$\frac{1}{2}b^{5}a$$