Give a proof by contradiction of each of the following state

Solution

Solution:

Suppose 8 = a/b with integers a, b

and gcd(a,b) = 1 (meaning the ratio is simplified)
then 8 = a²/b²
and 8b² = a²

this implies 8 divides a² which also means 8 divides a.

so there exists a p within the integers such that:
a = 8p
and thus,
8 = 8p/b
which implies
8 = 64p²/b²
which is:
1/8 = p²/b²
or:
b²/p² = 8
which implies
b² = 8p²

which implies 8 divides b² which means 8 divides b.

8 divides a, and 8 divides b,

which is a contradiction because gcd (a, b) = 1

therefore, the square root of 8 is irrational.