$(sqrt5-1)/(sqrt5+1)*[(1-4sqrt3)/2+(sqrt3-1)(sqrt3+1)-(sqrt3-1)^2]*4/(3-sqrt5)=$

A cura di: Francesca Ricci

$(sqrt5-1)/(sqrt5+1)*[(1-4sqrt3)/2+(sqrt3-1)(sqrt3+1)-(sqrt3-1)^2]*4/(3-sqrt5)=$

Svolgiamo le moltiplicazioni e il quadrato nella parentesi quadra:

$(sqrt5-1)/(sqrt5+1)*[(1-4sqrt3)/2+3-1-(3+1-2sqrt3)]*4/(3-sqrt5)=$

$(sqrt5-1)/(sqrt5+1)*[(1-4sqrt3)/2+3-1-3-1+2sqrt3]*4/(3-sqrt5)=$

$(sqrt5-1)/(sqrt5+1)*[(1-4sqrt3)/2-2+2sqrt3]*4/(3-sqrt5)=$

Svolgiamo il m.c.m. nella parentesi tonda:

$(sqrt5-1)/(sqrt5+1)*(1-4sqrt3-2*2+2sqrt3*2)/2*4/(3-sqrt5)=$

$(sqrt5-1)/(sqrt5+1)*(1-4sqrt3-4+4sqrt3)/2*4/(3-sqrt5)=$

$(sqrt5-1)/(sqrt5+1)*(-3/2)*4/(3-sqrt5)=$

Razionalizziamo:

$(sqrt5-1)/(sqrt5+1)*(sqrt5-1)/(sqrt5-1)(-3/2)*4/(3-sqrt5)*(3+sqrt5)/(3+sqrt5)=$

$((sqrt5-1)*(sqrt5-1))/((sqrt5+1)*(sqrt5-1))*(-3/2)*(4*(3+sqrt5))/((3-sqrt5)*(3+sqrt5))=$

$(sqrt5-1)^2/(5-1)*(-3/2)*(12+4sqrt5)/(9-5)=$

$(5+1-2sqrt5)/4*(-3/2)*(12+4sqrt5)/4=$

$(6-2sqrt5)/4*(-3/2)*(12+4sqrt5)/4=$

$(2(3-sqrt5))/4*(-3/2)*(4(3+sqrt5))/4=$

Moltiplichiamo le tre frazioni:

$(-3*(3-sqrt5)*(3+sqrt5))/(2*2)=$

$(-3*(9-5))/4=$

$(-3*4)/4=$

$-12/4=-3$