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

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    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)

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