Solve (1/5a^{2}-1/2ab)^{2}+1/2ab(a^2-1/2ab)-frac{3}{5}a^2(-a)^2-frac{3}{10}a(a^2b)

$$( \frac { 1 } { 5 } a ^ { 2 } - \frac { 1 } { 2 } a b ) ^ { 2 } + \frac { 1 } { 2 } a b ( a ^ { 2 } - \frac { 1 } { 2 } a b ) - \frac { 3 } { 5 } a ^ { 2 } ( - a ) ^ { 2 } - \frac { 3 } { 10 } a ( a ^ { 2 } b )$$

Evaluate

$$-\frac{14a^{4}}{25}$$

Solution Steps

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

$$\left(\frac{1}{5}a^{2}-\frac{1}{2}ab\right)^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(\frac{1}{5}a^{2}-\frac{1}{2}ab\right)^{2}.

$$\frac{1}{25}\left(a^{2}\right)^{2}-\frac{1}{5}a^{2}ab+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{2}ab+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{3}b+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{3}b+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ba^{3}-\frac{1}{4}a^{2}b^{2}-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Combine -\frac{1}{5}a^{3}b and \frac{1}{2}ba^{3} to get \frac{3}{10}a^{3}b.

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b+\frac{1}{4}a^{2}b^{2}-\frac{1}{4}a^{2}b^{2}-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Calculate -a to the power of 2 and get a^{2}.

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{2}a^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{4}-\frac{3}{10}a^{3}b$$

Combine \frac{1}{25}a^{4} and -\frac{3}{5}a^{4} to get -\frac{14}{25}a^{4}.

$$-\frac{14}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{10}a^{3}b$$

Combine \frac{3}{10}a^{3}b and -\frac{3}{10}a^{3}b to get 0.

$$-\frac{14}{25}a^{4}$$

Expand

$$-\frac{14a^{4}}{25}$$

Solution Steps

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

$$\left(\frac{1}{5}a^{2}-\frac{1}{2}ab\right)^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(\frac{1}{5}a^{2}-\frac{1}{2}ab\right)^{2}.

$$\frac{1}{25}\left(a^{2}\right)^{2}-\frac{1}{5}a^{2}ab+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{2}ab+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{3}b+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ab\left(a^{2}-\frac{1}{2}ab\right)-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}-\frac{1}{5}a^{3}b+\frac{1}{4}a^{2}b^{2}+\frac{1}{2}ba^{3}-\frac{1}{4}a^{2}b^{2}-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Combine -\frac{1}{5}a^{3}b and \frac{1}{2}ba^{3} to get \frac{3}{10}a^{3}b.

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b+\frac{1}{4}a^{2}b^{2}-\frac{1}{4}a^{2}b^{2}-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{2}\left(-a\right)^{2}-\frac{3}{10}a^{3}b$$

Calculate -a to the power of 2 and get a^{2}.

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{2}a^{2}-\frac{3}{10}a^{3}b$$

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

$$\frac{1}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{5}a^{4}-\frac{3}{10}a^{3}b$$

Combine \frac{1}{25}a^{4} and -\frac{3}{5}a^{4} to get -\frac{14}{25}a^{4}.

$$-\frac{14}{25}a^{4}+\frac{3}{10}a^{3}b-\frac{3}{10}a^{3}b$$

Combine \frac{3}{10}a^{3}b and -\frac{3}{10}a^{3}b to get 0.

$$-\frac{14}{25}a^{4}$$