124/MR00. problem / Math - Solved Math Problems

MR-124. problem

Simplify the following expression:

$\frac{2 x^{n} - 2 x^{- n}}{2 x^{n} + 2 x^{- n} + 4}$

$x \neq 0; x \neq - 1$

$\frac{2 x^{n} - 2 x^{- n}}{2 x^{n} + 4 x^{- n} + 4}$

$\frac{2 (x^{n} - x^{- n})}{2 (x^{n} + x^{- n} + 2)}$

$= \frac{x^{n} - \frac{1}{x^{n}}}{x^{n} + \frac{1}{x^{n}} + 2}$

$= \frac{\frac{x^{n} \cdot x^{n} - 1}{x^{n}}}{\frac{x^{n} \cdot x^{n} + 1 + 2 x^{n}}{x^{n}}}$

$= \frac{\frac{x^{2 n} - 1}{x^{n}}}{\frac{x^{2 n} + 2 x^{n} + 1}{x^{n}}}$

$= \frac{x^{n} (x^{2 n} - 1)}{x^{n} (x^{2 n} + 2 x^{n} + 1)}$

$= \frac{x^{2 n} - 1}{x^{2 n} + 2 x^{n} + 1}$

$= \frac{(x^{n} - 1) (x^{n} + 1)}{{(x^{n} + 1)}^{2}}$

$= \frac{x^{n} - 1}{x^{n} + 1}$

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