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    show that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers

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    Show that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.

    © BrainMass Inc. brainmass.com April 3, 2020, 2:55 pm ad1c9bdddf
    https://brainmass.com/math/fractions-and-percentages/show-that-2-1-3-5-1-7-and-13-1-4-do-not-represent-rational-numbers-50559

    Solution Preview

    We will prove this by contradiction.

    The idea of a proof by contradiction is to:

    First, we assume that the opposite of what we wish to prove is true.
    Then, we show that the logical consequences of the assumption include a contradiction.
    Finally, we conclude that the assumption must have been false

    Recall, A number r is rational if it can be written as a fraction r = p/q where both p and q are integers.

    Proof: For 2^1/3

    (If 2^1/3 is rational, it should be representable as a fraction r = p/q)

    Assume that a rational number r exists such that r^3 = 2. (or, r = 2^1/3)

    As we have already assumed that, r is rational, in the representation r=p/q assume p and q are mutually prime, i.e. have no common divisors. The fraction p/q in this case is called irreducible. In other words, a/b is ...

    Solution Summary

    It is shown that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.

    $2.19

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