Solve (3/4n^{2}-1/6n)^{2}-7/4(1/2n^2-n)(frac{1}{2}n^2+n)-(frac{1}{2}n^2-frac{1}{3}n)(frac{1}{4}n^2-frac{1}{3}n)

$$( \frac { 3 } { 4 } n ^ { 2 } - \frac { 1 } { 6 } n ) ^ { 2 } - \frac { 7 } { 4 } ( \frac { 1 } { 2 } n ^ { 2 } - n ) ( \frac { 1 } { 2 } n ^ { 2 } + n ) - ( \frac { 1 } { 2 } n ^ { 2 } - \frac { 1 } { 3 } n ) ( \frac { 1 } { 4 } n ^ { 2 } - \frac { 1 } { 3 } n )$$

Evaluate

$$\frac{5n^{2}}{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}n^{2}-\frac{1}{6}n\right)^{2}.

$$\frac{9}{16}\left(n^{2}\right)^{2}-\frac{1}{4}n^{2}n+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{2}n+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{9}n^{2}\right)$$

To find the opposite of \frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{9}n^{2}, find the opposite of each term.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Use the distributive property to multiply -\frac{7}{4} by \frac{1}{2}n^{2}-n.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}+\left(-\frac{7}{8}n^{2}+\frac{7}{4}n\right)\left(\frac{1}{2}n^{2}+n\right)-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Use the distributive property to multiply -\frac{7}{8}n^{2}+\frac{7}{4}n by \frac{1}{2}n^{2}+n and combine like terms.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{16}n^{4}+\frac{7}{4}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{9}{16}n^{4} and -\frac{7}{16}n^{4} to get \frac{1}{8}n^{4}.

$$\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}+\frac{7}{4}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{1}{36}n^{2} and \frac{7}{4}n^{2} to get \frac{16}{9}n^{2}.

$$\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{16}{9}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{1}{8}n^{4} and -\frac{1}{8}n^{4} to get 0.

$$-\frac{1}{4}n^{3}+\frac{16}{9}n^{2}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine -\frac{1}{4}n^{3} and \frac{1}{4}n^{3} to get 0.

$$\frac{16}{9}n^{2}-\frac{1}{9}n^{2}$$

Combine \frac{16}{9}n^{2} and -\frac{1}{9}n^{2} to get \frac{5}{3}n^{2}.

$$\frac{5}{3}n^{2}$$

Expand

$$\frac{5n^{2}}{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}n^{2}-\frac{1}{6}n\right)^{2}.

$$\frac{9}{16}\left(n^{2}\right)^{2}-\frac{1}{4}n^{2}n+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{2}n+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{2}n^{2}-\frac{1}{3}n\right)\left(\frac{1}{4}n^{2}-\frac{1}{3}n\right)$$

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

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\left(\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{9}n^{2}\right)$$

To find the opposite of \frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{9}n^{2}, find the opposite of each term.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{4}\left(\frac{1}{2}n^{2}-n\right)\left(\frac{1}{2}n^{2}+n\right)-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Use the distributive property to multiply -\frac{7}{4} by \frac{1}{2}n^{2}-n.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}+\left(-\frac{7}{8}n^{2}+\frac{7}{4}n\right)\left(\frac{1}{2}n^{2}+n\right)-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Use the distributive property to multiply -\frac{7}{8}n^{2}+\frac{7}{4}n by \frac{1}{2}n^{2}+n and combine like terms.

$$\frac{9}{16}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}-\frac{7}{16}n^{4}+\frac{7}{4}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{9}{16}n^{4} and -\frac{7}{16}n^{4} to get \frac{1}{8}n^{4}.

$$\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{1}{36}n^{2}+\frac{7}{4}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{1}{36}n^{2} and \frac{7}{4}n^{2} to get \frac{16}{9}n^{2}.

$$\frac{1}{8}n^{4}-\frac{1}{4}n^{3}+\frac{16}{9}n^{2}-\frac{1}{8}n^{4}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine \frac{1}{8}n^{4} and -\frac{1}{8}n^{4} to get 0.

$$-\frac{1}{4}n^{3}+\frac{16}{9}n^{2}+\frac{1}{4}n^{3}-\frac{1}{9}n^{2}$$

Combine -\frac{1}{4}n^{3} and \frac{1}{4}n^{3} to get 0.

$$\frac{16}{9}n^{2}-\frac{1}{9}n^{2}$$

Combine \frac{16}{9}n^{2} and -\frac{1}{9}n^{2} to get \frac{5}{3}n^{2}.

$$\frac{5}{3}n^{2}$$