Prove that √5 is irrational number.if …

Let's prove this by the method of contradiction-
Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers.
⇒√5=p/q
⇒5=p²/q²  {Squaring both the sides}
⇒5q²=p²  (1)
⇒p² is a multiple of 5. {Euclid's Division Lemma}
⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}
⇒p=5m
⇒p²=25m²   (2)
From equations (1) and (2), we get,
5q²=25m²
⇒q²=5m²
⇒q² is a multiple of 5. {Euclid's Division Lemma}
⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}
Hence, p,q have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number. 
For you second query, as we've proved √5 irrational. Therefore √5+3 is also irrational because sum of a rational and an irrational number is always an irrational number.