121/MR00. zadatak / Matematika - Zbirka rešenih zadataka

MR-121. zadatak

Uprostite sledeći izraz:

$2 x^{- n} {(x^{n} - 1)}^{- 1} - 4 {(x^{2 n} - 1)}^{- 1} + 2 x^{- n} {(x^{n} + 1)}^{- 1}$

$x \neq 0$

$2 x^{- n} {(x^{n} - 1)}^{- 1} - 4 {(x^{2 n} - 1)}^{- 1} + 2 x^{- n} {(x^{n} + 1)}^{- 1}$

$= \frac{2}{x^{n} (x^{n} - 1)} - \frac{4}{(x^{n} - 1) (x^{n} + 1)} + \frac{2}{x^{n} (x^{n} + 1)}$

$= \frac{2 (x^{n} + 1) - 4 x^{n} + 2 (x^{n} - 1)}{x^{n} (x^{n} - 1) (x^{n} + 1)}$

$= \frac{2 x^{n} + 2 - 4 x^{n} + 2 x^{n} - 2}{x^{n} (x^{n} - 1) (x^{n} + 1)}$

$= \frac{4 x^{n} - 4 x^{n} + 2 - 2}{x^{n} (x^{n} - 1) (x^{n} + 1)}$

$= \frac{0}{x^{n} (x^{n} - 1) (x^{n} + 1)}$

$= 0$

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