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Regarding the proof that 8^sqrt(7) < 7^sqrt(8)
Posted:
Jun 16, 1996 9:31 PM


Everyone:
I have spent a considerable amount of time attempting to derive the inequality 8^sqrt(7) < 7^sqrt(8) and have gotten tantalizingly close, but not completed the thing.
I have seen a few "proofs," but have not understood them in their entirety, particularly:
"It suffices to prove that
8^sqrt(7) < 7^[31 sqrt(7) / 29]
or
8 / 7 < 7^30 / 8^28."
How on Earth do these two inequalities relate? I would appreciate a brief explanation and/or some references. I do admit that inequalities are a weak spot where I am concerned. I have no real "feel" for them. I tried to correct that this year when I developed several of them for a small project in Real Analysis. Still, I am not comfortable with them.
Wait. . . nevermind. <Grin> I believe it goes like this:
8^sqrt(7) < 7^[31 sqrt(7) / 29] 8^[29 sqrt(7)] < 7^[31 sqrt(7)] 8^29 < 7^31 8/7 < 7^30/8^28
Aha. I wanted to post this anyhow  just in case I was the only one clouded up about this.
(Ahem) As you were. . . .
J. B. Rainsberger,  Any sixyearold who knows the York University  words nefarious and extemp (collÃÂÃÂ¨ge Glendon)  oraneous gets my vote. t762@inforamp.net   Joe on Calvin.



