Question

please solve this question .please​

Answer 1

Given a + √b = c + √d Case (i): Let a=c ⇒ a + √b = c + √d becomes a + √b = a + √d ⇒ √b = √d ∴ b = d Case (ii): Let a ≠ c Let us take a = c + k where k is a rational number not equal to zero. ⇒ a + √b = c + √d becomes (c + k) + √b = c + √d ⇒ k + √b = √d Let us now square on both the sides, ⇒ (k + √b)2 = (√d)2 ⇒ k2 + b + 2k√b = d ⇒ 2k√b = d – k2 – b Notice that the RHS is a rational number. Hence √b is a rational number This is possible only when b is square of a rational number. Thus d is also square of a rational number as k + √b = √d.