Question

Let a, b, c, d be positive rationales such that

$a \: + \sqrt{b} = \: c + \: \sqrt{d}$

Then show that either a=c and b=d or b and d are square of rationals.

❌Don't answer if you don't know❌

BEST ANSWER WILL BE MARKED AS BRAINLIEST

Answer 1

Answer:

b and d are square of rationals!

Step-by-step explanation:

(i)

Let a = c:

Given: a + √b = c + √d.

⇒ a + √b = a + √d

⇒ √b = √d

On Squaring both sides, we get

⇒ b = d.

(ii)

Let a ≠ c and a = c + p{p is some rational number}

⇒ a + √b = c + √d

⇒ c + p + √b = c + √d

⇒ p + √b = √d

On Squaring both sides, we get

⇒ (p + √b)² = √d)²

⇒ p² + b + 2p√b = d

⇒ 2p√b = d - p² - b

⇒ √b = (d - p² - b/2p]

∴ Hence √b is a rational number.

This is possible only when b is a square of rational number.

Hence, d is also a square of rational number .

Hope it helps!