Prove that √5 is irrational number.if it is so,then prove 3+√5 is also a irrational number.

Prove that √5 is irrational number.if it is so,then prove 3+√5 is

1.	Prove that √5 is irrational number.if it is so,then prove 3+√5 is also a irrational number.
Answer» Let\'s prove this by the method of contradiction-Say,\xa0√5 is a rational number.\xa0∴ 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\xa0√5 is an irrational number.\xa0For you second query, as we\'ve proved\xa0√5 irrational. Therefore\xa0√5+3 is also irrational because sum of a rational and an irrational number is always an irrational number.