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.