Question

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Answer 1

Proof :

Let us take (√2 + √7) as a rational number. Then,

√2 + √7 = a/b, where both a and b are integers with non-zero b

⇒ √2 = a/b - √7

Squaring both sides, we get

(√2)² = (a/b - √7)²

⇒ 2 = a²/b² - 2 (a/b) √7 + 7

⇒ 2 (a/b) √7 = a²/b² + 7 - 2

⇒ 2 (a/b) √7 = a²/b² + 5

⇒ 2√7 = a/b + 5b/a, multiplying both sides by b/a

⇒ √7 = a/(2b) + 5b/(2a)

This shows that the right hand side a/(2b) + 5b/(2a) is a rational number since both and b are integers, but this leads to a contradiction to the fact that √7 is an irrational number.

So, our assumption is wrong.

Therefore, (√2 + √7) is not a rational number.

Hence, proved.