Question

prove that √2 + √7 is not rational no.

Answer 1

Let √2+√7 be a rational number.
A rational number can be written in the form of p/q.
√2+√7=p/q

Squaring on both sides,
(√2+√7)²=(p/q)²
[√2²+√7²+2(√2)(√7)]=p²/q²
(2+7+2√14)=p²/q²
(2√14+9)=p²/q²
2√14=p²/q²-9
2√14=(p²-9q²)/q²
√14=(p²-9q²)/2q²

p,q are integers then (p²-9q²)/2q² is a rational number.
Then √14 is also a rational number.
But this contradicts the fact that √14 is an irrational number.
So,our supposition is false.
Therefore,√2+√7 is an irrational number.
Hence proved.