Frazioni algebriche – Esercizio 49

Frazioni algebriche

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Esercizio.  (\bigstar\bigstar\bigstar)

Semplificare la seguente espressione assumendo che siano verificate le condizioni di esistenza:

    \[\left\{ \left[\left(1-\dfrac{1}{x}\right):\left(1-\dfrac{1}{x^2}\right)\right]^2-\dfrac{x^2}{x^2+2x+1}\right\}:\dfrac{1}{x^2-1}\]

 

Soluzione. 
Sfruttando

    \[A^2-B^2 = (A-B)(A+B)\]

e

    \[A^2+2AB+B^2=(A+B)^2\]

possiamo scrivere

    \[\begin{aligned} & \left\{ \left[\left(1-\dfrac{1}{x}\right):\left(1-\dfrac{1}{x^2}\right)\right]^2-\dfrac{x^2}{x^2+2x+1}\right\}:\dfrac{1}{x^2-1} =\\\\ & = \left\{ \left[\left(\dfrac{x-1}{x}\right):\left(\dfrac{(x-1)(x+1)}{x^2}\right)\right]^2-\dfrac{x^2}{(x+1)^2}\right\}:\dfrac{1}{(x-1)(x+1)} =\\\\ & = \left\{ \dfrac{x^2}{(x+1)^2} -\dfrac{x^2}{(x+1)^2}\right\}:\dfrac{1}{(x-1)(x+1)} =\\\\ & = 0:\dfrac{1}{(x-1)(x+1)} =\\\\ & = 0 \end{aligned}\]

 


Fonte: Moduli di lineamenti di matematica N.Dodero – P.Baroncini – R.Manfredi