Question

prove that root 7 is irrational​

Answer 1

Step-by-step explanation:

Let us assume that 7 is rational. Then, there exist co-prime positive integers a and b such that

7=ba

⟹a=b7

Squaring on both sides, we get

a2=7b2

Therefore, a2 is divisible by 7 and hence, a is also divisible by7

so, we can write a=7p, for some integer p.

Substituting for a, we get 49p2=7b2⟹b2=7p2.

This means, b2 is also divisible by 7 and so, b is also divisible by 7.

Therefore, a and b have at least one common factor, i.e., 7.

But, this contradicts the fact that a and b are co-prime.

Thus, our supposition is wrong.

Hence, 7 is irrational