Solve 1/16(4/3a-1)(4/3a+1)(16/9a^2+1)+frac{1}{2}(frac{4}{9}a^2+frac{1}{4})-(frac{4}{9}a^2+frac{1}{4})^2

$$\frac { 1 } { 16 } ( \frac { 4 } { 3 } a - 1 ) ( \frac { 4 } { 3 } a + 1 ) ( \frac { 16 } { 9 } a ^ { 2 } + 1 ) + \frac { 1 } { 2 } ( \frac { 4 } { 9 } a ^ { 2 } + \frac { 1 } { 4 } ) - ( \frac { 4 } { 9 } a ^ { 2 } + \frac { 1 } { 4 } ) ^ { 2 }$$

Evaluate

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Solution Steps

Use the distributive property to multiply \frac{1}{16} by \frac{4}{3}a-1.

$$\left(\frac{1}{12}a-\frac{1}{16}\right)\left(\frac{4}{3}a+1\right)\left(\frac{16}{9}a^{2}+1\right)+\frac{1}{2}\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)-\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}$$

Use the distributive property to multiply \frac{1}{12}a-\frac{1}{16} by \frac{4}{3}a+1 and combine like terms.

$$\left(\frac{1}{9}a^{2}-\frac{1}{16}\right)\left(\frac{16}{9}a^{2}+1\right)+\frac{1}{2}\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)-\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}$$

Use the distributive property to multiply \frac{1}{9}a^{2}-\frac{1}{16} by \frac{16}{9}a^{2}+1 and combine like terms.

$$\frac{16}{81}a^{4}-\frac{1}{16}+\frac{1}{2}\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)-\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}$$

Use the distributive property to multiply \frac{1}{2} by \frac{4}{9}a^{2}+\frac{1}{4}.

$$\frac{16}{81}a^{4}-\frac{1}{16}+\frac{2}{9}a^{2}+\frac{1}{8}-\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}$$

Add -\frac{1}{16} and \frac{1}{8} to get \frac{1}{16}.

$$\frac{16}{81}a^{4}+\frac{1}{16}+\frac{2}{9}a^{2}-\left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(\frac{4}{9}a^{2}+\frac{1}{4}\right)^{2}.

$$\frac{16}{81}a^{4}+\frac{1}{16}+\frac{2}{9}a^{2}-\left(\frac{16}{81}\left(a^{2}\right)^{2}+\frac{2}{9}a^{2}+\frac{1}{16}\right)$$

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

$$\frac{16}{81}a^{4}+\frac{1}{16}+\frac{2}{9}a^{2}-\left(\frac{16}{81}a^{4}+\frac{2}{9}a^{2}+\frac{1}{16}\right)$$

To find the opposite of \frac{16}{81}a^{4}+\frac{2}{9}a^{2}+\frac{1}{16}, find the opposite of each term.

$$\frac{16}{81}a^{4}+\frac{1}{16}+\frac{2}{9}a^{2}-\frac{16}{81}a^{4}-\frac{2}{9}a^{2}-\frac{1}{16}$$

Combine \frac{16}{81}a^{4} and -\frac{16}{81}a^{4} to get 0.

$$\frac{1}{16}+\frac{2}{9}a^{2}-\frac{2}{9}a^{2}-\frac{1}{16}$$

Combine \frac{2}{9}a^{2} and -\frac{2}{9}a^{2} to get 0.

$$\frac{1}{16}-\frac{1}{16}$$

Subtract \frac{1}{16} from \frac{1}{16} to get 0.

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Factor

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Solution Steps

Factor out common term 16a^{2}+9 by using distributive property.

$$\left(16a^{2}+9\right)\left(-\frac{1}{144}+\frac{1}{81}a^{2}+\frac{1}{72}-\frac{1}{144}-\frac{1}{81}a^{2}\right)$$

Consider -\frac{1}{144}+\frac{1}{81}a^{2}+\frac{1}{72}-\frac{1}{144}-\frac{1}{81}a^{2}. Simplify.

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