Question: Simplify: $\dfrac{b}{a-b} +$$\dfrac{b}{a+b} +$$\dfrac{2ab}{a²+b²} +$$\dfrac{4a³b}{a⁴+b⁴}$

Solution:
Given,

$= \dfrac{b}{a-b} +\dfrac{b}{a+b} +\dfrac{2ab}{a²+b²} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{b(a+b} +b(a-b)}{(a-b)(a+b)} +\dfrac{2ab}{a²+b²} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{ab +b² +ab -b²}{a² -b²} +\dfrac{2ab}{a²+b²} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{2ab}{a²-b²} +\dfrac{2ab}{a²+b²} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{2ab(a²+b²) +2ab(a²-b²)}{(a²-b²)(a²+b²)} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{2a³b +2ab³ +2a³b -2ab³}{a⁴ -b⁴} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{4a³b}{a⁴ -b⁴} +\dfrac{4a³b}{a⁴+b⁴}$

$= \dfrac{4a³b(a⁴ +b⁴) + 4a³b(a⁴ -b⁴)}{(a⁴ -b⁴)(a⁴ +b⁴)}$

$= \dfrac{4a⁷b +4a³b⁵ +4a⁷b -4a³b⁵}{a⁸ -b⁸}$

$= \dfrac{8a⁷b}{a⁸ - b⁸}$
= Answer

Related Notes and Solutions:

Here is the Website link to the guide of Simplification of Rational Expressions.

Here is the Page link to all the solutions of Simplification of Rational Expressions.

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