# Explain the Proof Step-by-step : If n is not a perfect square, then n^(1/2) is irrational.

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k.

Theorem.

If the natural number n is not a perfect square, then n^(1/2) is irrational.

Proof.

S(1):

Suppose n^(1/2) = r/s for some natural numbers r and s.

S(2):

We may assume that r and s have no prime factors in common, since any

common prime factors may be cancelled.

S(3):

From the first step, we have (s^2)*n = r^2.

S(4):

Suppose that s>1 and p is a prime factor of s.

S(5):

Then p is a prime factor of s^2.

S(6):

Hence p is a prime factor of r^2 = (s^2)*n.

S(7):

It follows that p is a prime factor of r.

S(8):

This contradicts our assumption that r and s have no prime factors in

common, and so s = 1.

S(9):

Therefore, n = (r^2), so n is a perfect square.

Explain all S(1) ~ S(9)

Please see the attached file for the fully formatted problems.

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