Question

prove that 1/2+root 3 +2/root 5-root 3 +1/2-root 5=0​

Answer 1

LHS

= 1 / (2 + √3) + 2 / (√5 - √3) + 1 / (2 - √5)

= (2 - √3) / [(2 + √3)(2 - √3)] + 2(√5 + √3) / [(√5 - √3)(√5 + √3)] + (2 + √5) / [(2 - √5)(2 + √5)]

= (2 - √3) / (4 - 3) + (2√5 + 2√3) / (5 - 3) + (2 + √5) / (4 - 5)

= [(2 - √3) / 1] + [2(√5 + √3) / 2] + [(2 + √5) / -1]

= 2 - √3 + √5 + √3 - 2 - √5

= 0

= RHS

LHS = RHS

Hence Proved!

Hope this helps! :)