$sqrt(a^2/5-1+4/(5a^2)):(sqrt(a-3+2/a)sqrt((a^2+a)/(5a+10)))=$

A cura di: Francesco Speciale

Risolvere la seguente espressione
$sqrt(a^2/5-1+4/(5a^2)):(sqrt(a-3+2/a)sqrt((a^2+a)/(5a+10)))=$

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$sqrt(a^2/5-1+4/(5a^2)):(sqrt(a-3+2/a)sqrt((a^2+a)/(5a+10)))=$
$=sqrt((a^4-5a^2+4)/(5a^2)):(sqrt((a^2-3a+2)/a)sqrt((a(a+1))/(5(a+2))))=$
Essendo $a^4-5a^2+4=(a^2-4)(a^2-1)$ e $a^2-3a+2=(a-2)(a-1)$ si ha
$=sqrt((a^4-5a^2+4)/(5a^2)):(sqrt((a^2-3a+2)/a)sqrt((a(a+1))/(5(a+2))))=$
$=sqrt(((a^2-4)(a^2-1))/(5a^2)):(sqrt(((a-2)(a-1))/a)sqrt((a(a+1))/(5(a+2))))=$
Semplificando
$=sqrt(((a^2-4)(a^2-1))/(5a^2)):(sqrt(((a-2)(a-1)(a+1))/(5(a+2)))=$
$=sqrt(((a-2)(a+2)(a-1)(a+1))/(5a^2)(5(a+2))/((a-2)(a-1)(a+1)))=$
Semplificando ancora si ottiene
$=sqrt((a+2)^2/(a^2))=|(a+2)/a|$.