Solve 1/a-2b+1/a+2b+2a/a^{2+4b^2}+4a^3/a^4+16b^4

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

Evaluate

$$\frac{8a^{7}}{a^{8}-256b^{8}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2b and a+2b is \left(a-2b\right)\left(a+2b\right). Multiply \frac{1}{a-2b} times \frac{a+2b}{a+2b}. Multiply \frac{1}{a+2b} times \frac{a-2b}{a-2b}.

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

Since \frac{a+2b}{\left(a-2b\right)\left(a+2b\right)} and \frac{a-2b}{\left(a-2b\right)\left(a+2b\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in a+2b+a-2b.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2b\right)\left(a+2b\right) and a^{2}+4b^{2} is \left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right). Multiply \frac{2a}{\left(a-2b\right)\left(a+2b\right)} times \frac{a^{2}+4b^{2}}{a^{2}+4b^{2}}. Multiply \frac{2a}{a^{2}+4b^{2}} times \frac{\left(a-2b\right)\left(a+2b\right)}{\left(a-2b\right)\left(a+2b\right)}.

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

Since \frac{2a\left(a^{2}+4b^{2}\right)}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)} and \frac{2a\left(a-2b\right)\left(a+2b\right)}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2a\left(a^{2}+4b^{2}\right)+2a\left(a-2b\right)\left(a+2b\right).

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

Combine like terms in 2a^{3}+8ab^{2}+2a^{3}+4a^{2}b-4a^{2}b-8ab^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right) and a^{4}+16b^{4} is \left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)\left(a^{4}+16b^{4}\right). Multiply \frac{4a^{3}}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)} times \frac{a^{4}+16b^{4}}{a^{4}+16b^{4}}. Multiply \frac{4a^{3}}{a^{4}+16b^{4}} times \frac{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)}.

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

Since \frac{4a^{3}\left(a^{4}+16b^{4}\right)}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)\left(a^{4}+16b^{4}\right)} and \frac{4a^{3}\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)\left(a^{4}+16b^{4}\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4a^{3}\left(a^{4}+16b^{4}\right)+4a^{3}\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right).

$$\frac{4a^{7}+64a^{3}b^{4}+4a^{7}+16a^{5}b^{2}+8a^{6}b+32a^{4}b^{3}-8a^{6}b-32a^{4}b^{3}-16a^{5}b^{2}-64a^{3}b^{4}}{\left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)\left(a^{4}+16b^{4}\right)}$$

Combine like terms in 4a^{7}+64a^{3}b^{4}+4a^{7}+16a^{5}b^{2}+8a^{6}b+32a^{4}b^{3}-8a^{6}b-32a^{4}b^{3}-16a^{5}b^{2}-64a^{3}b^{4}.

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

Expand \left(a-2b\right)\left(a+2b\right)\left(a^{2}+4b^{2}\right)\left(a^{4}+16b^{4}\right).

$$\frac{8a^{7}}{a^{8}-256b^{8}}$$

Differentiate w.r.t. a

$$\frac{8a^{6}\left(-a^{8}-1792b^{8}\right)}{\left(a^{8}-256b^{8}\right)^{2}}$$