Question

Prove that, if A and B and C and D are positive nationals such that a + b root b equal to 3 + root 3 then either equal to C and B equal to D or b and D are squares of rationals

Answer 1

Answer:

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