Question

Prove that root 3+root 5 is irrational.

Answer 1

Let √3+√5=r where r is a rational number

=>(√3+√5)^2 =r^2

=>3+5+2√15=r^2

=>2√15=r^2-8

=>√15=(r^2-8)/2

√15 is irrational but (r^2 -8)/2 is irrational

here arised a contradiction due to our incorrect assumption that √3+√5 is rational

hence √3+√5 is irrational

Answer 2

Let us suppose that √3+√5 is rational.

Let √3+√5 = a , where is rational

Therefore,   √3 = a-√5

On squaring both sides , we get

(√3)² = (a-√5)²

3 = a²+5-2a√5

2a√5 = a²+2

√5 = a²+2/2a which is contradiction.

As the right hand is rational number while √5 is irrational.Since 3 and 5 are prime numbers.Hence √3+√5 is irrational