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Thread: Which side is larger?

  1. #1
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    Which side is larger?

    Determine which side is larger without using a calculator
    or computer with showing your steps.

    $81^{81.25} \ \ vs. \ \ 82^{81}$
    Thanks from topsquark
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  2. #2
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    Re: Which side is larger?

    Spoiler:

    $81^{81.25} = 3\cdot 81^{81}$

    $82^{81} =81^{81} \left(\dfrac{1+81}{81}\right)^{81}$

    $\lim_{n \to \infty} ~\left(\dfrac{1+n}{n}\right)^n = e$

    and this limit is approached strictly from below.

    Thus $\left(\dfrac{1+81}{81}\right)^{81} < e < 3$

    So comparing the two expressions we have

    $3 \cdot 81^{81} > \left(\dfrac{1+81}{81}\right)^{81} \cdot 81^{81}$

    and thus $81^{81.25} > 82^{81}$
    Last edited by romsek; Apr 27th 2018 at 11:11 PM.
    Thanks from greg1313, topsquark and SlipEternal
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