Prove that root 8 is irrational number

1.	Prove that root 8 is irrational number
Answer» suppose √8 = a/b with integers a, band gcd(a,b) = 1 (meaning the ratio is simplified)then 8 = a²/b²and 8b² = a²this implies 8 divides a² which also means 8 divides a.so there exists a p within the integers such that:a = 8pand thus,√8 = 8p/bwhich implies8 = 64p²/b²which is:1/8 = p²/b²or:b²/p² = 8which impliesb² = 8p²which implies 8 divides b² which means 8 divides b.8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1therefore, the square root of 8 is irrational.

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