Question

Prove that root 3 +root5 is irrational

Answer 1

let us suppose 3+root 5 is rational.
=>3+root 5 is in the form of p/q where p and q are integers and q is not =0
=>root5=p/q-3
​=>root 5=p-3q/q
as p, q and 3 are integers p-3q/3 is a rational number.
=>root 5 is a rational number.
but we know that root 5 is an irrational number.
this is an contradiction.
this contradiction has arisen because of our wrong assumption that 3+root 5 is a rational number.
hence 3+ root 5 is an irrational number.

Answer 2

Hello friend

Here's your answer

To prove: √3+√5 is irrational

To prove it let us assume it to be a rational number

Rational numbers are the ones which can be expressed in p/q form where p,q are integers and q isn't equal to 0

√3+√5=p/q

√3=(p/q)-√5

Squatting on both sides

3=p²/q²-(2√5p)/q+5

(2√5p)/q=5-3-p²/q²

2√5p/q=(2q²-p²)/q²

√5=(2q²-p²)/q²*q/2p

√5=(2q²-p²)/2pq

As p and q are integers RHS is rational

As RHS is rational LHS is also rational i.e √5 is rational

But this contradicts the fact that √5 is irrational

This contradiction arose because of our false assumption

So √3+√5 is irrational.

Hope it helps!!!