Question

solved the question no ( 1 )√3-1÷√3+1=a+b√3

Answer 1

just rationalize, means multiply both the denominator and numerator by (√3 - 1).
and now your question simplifies to:
{(√3-1) × (√3-1)} / {(√3+1) × (√3-1)}
→ (√3-1)² / (√3)²-(1)² = (3 + 1 - 2√3) / (3-1)
= (4 - 2√3) / (2)
= (2 - √3)
= a + b√3
on comparing we get
a = 2
b = -1

Answer 2

Answer :-

     (√3 - 1) / √3 + 1 = a + b√3

           ⇒  √3 - 1 (√3 - 1) / √3 + 1 (√3 - 1)

          ⇒  (√3 - 1)² / (√3)² - 1²

          ⇒  [(√3)² - 2 × √3 × 1 + 1²] / 3 - 1

          ⇒  [3 - 2√3 + 1] / 2

          ⇒  [4 - 2√3] / 2

          ⇒  2(2 - 2√3) / 2

        ⇒ 2 - √3

∴ a = 2

  b = -1

               HOPE IT HELPS ...