prove that the difference and quotient of 5 + 2 root 3 and 5 minus 2 root 3 irrational?

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  • 1 month ago

    If you mean:

    (5 + 2√3) - (5 - 2√3)

    Distribute the -1 to get:

    5 + 2√3 - 5 + 2√3

    4√3

    That's irrational.

    If dividing:

    (5 + 2√3) / (5 - 2√3)

    Rationalize the denominator by multiplying both halves by the conjugate of the denominator:

    (5 + 2√3)² / [(5 - 2√3)(5 + 2√3)]

    (25 + 20√3 + 4 * 3) / (25 - 4 * 3)

    (25 + 20√3 + 12) / (25 - 12)

    (37 + 20√3) / 13

    or:

    37/13 + (20/13)√3

    And that's still irrational.

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