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Show that the product of two of the numbers $(65^{1000} - 8^{2001} + 3^{177}), (79^{1212} - 9^{2399} + 2^{2001})$ and $(24^{4493} - 5^{8192} + 7^{1777})$ is non-negative, without actually evaluating the numbers.

P.S. I have found by calculation that all the three numbers are positive, but that does not solve the problem of proving without calculation.

Thanks in advance.

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    What is the goal of your question? What type of solution do you want? Intuitively, in any of the given parentheses the largest base is substantially larger than the others and with large exponents so one would think that the answer would be positive.2011-11-09
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    No, not intuitively, a more rigorous proof would be appreciated. And the smaller bases have larger powers, it's not so obvious that the numbers are positive.2011-11-09
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    What do you mean by "rigorous proof?" Are you just wanting a heuristic approach to solving the problem without the actual calculation?2011-11-09

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