Question

Prove that √3+√5 is irrational.....PROVE!!!!!

Answer 1

Hi there!

Let √3+√5 be a rational number.
A rational number can be written in the form of p/q where p,q are integers.

√3+√5 = p/q

√3 = p/q-√5

Squaring on both sides,

(√3)² = (p/q-√5)²

3 = p²/q²+√5²-2(p/q)(√5)

√5×2p/q = p²/q²+5-3

√5 = (p²+2q²)/q² × q/2p

√5 = (p²+2q²)/2pq

p,q are integers then (p²+2q²)/2pq is a rational number.
Then √5 is also a rational number.

But this contradicts the fact that √5 is an irrational number.
So,our supposition is false.

Therefore, √3+√5 is an irrational number.

[ Thank you! for asking the question. ]
Hope it helps!