Question

Prove that √2 + √5 is irrational.
Pls answer

Answer 1

Let √2+√5 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

√2+√5 = p/q

Squaring on both sides,

(√2+√5)² = (p/q)²

√2²+√5²+2(√5)(√2) = p²/q²

2+5+2√10 = p²/q²

7+2√10 = p²/q²

2√10 = p²/q² - 7

√10 = (p²-7q²)/2q

p,q are integers then (p²-7q²)/2q is a rational number.

Then √10 is also a rational number.

But this contradicts the fact that √10 is an irrational number.

.°. Our supposition is false.

√2+√5 is an irrational number.

Hence proved.

Answer 2

let √2+√5=p/q(rational)
(√2+√5)^2=p^2/q^2
2+5+2√10=p^2/q^2
√10=p^2/q^2-7/2
here lhs is irrational but rhs is rational
so.......