Question

Prove that root 5 + root 3 is not a rational number

Answer 1

Assume that √3 + √5 = p/q (it's rational). 

Multiple both sides by (√5 - √3). 

(√5 - √3) (√5 + √3) = 5-3 = 2 = p/q * (√5 - √3) 

(√5 - √3) = 2q/p, therefore √5 - √3 is rational = 2q/p 

√5 + √3 = p/q 
√5 - √3 = 2q/p 
√5 = [(p/q) + (2q/p)]/2, a rational number. 

But we know that √5 is IRRATIONAL (easily provable, let me know if you need the proof). 

Therefore the assumption is wrong and √3 + √5 is irrational. 

Answer 2

let √5+√3 be rational
⇒it can be written in the form p/q where p and q are co-primes and q≠0
⇒√5+√3=p/q
⇒√5=p/q-√3
squaring on both sides
(√5)²=(p/q-√3)²
⇒5=p²/q²-2√3p/q+3
⇒2=p²/q²-2√3p/q
⇒2-p²/q²=2√3p/q
⇒p²-2q²/q²=2p/q√3
⇒p²-2q²/2pq=√3
since p and q are integers therefore p²-2q²/2pq are rational but √3 is irrational(prove is there if you want)
⇒it is a contradiction
⇒√5+√3 is irrational