Order the given radical numbers

Thread starter littlemathquark
Start date Feb 18, 2025
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Arithmetic mean Radicals

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SUMMARY

The discussion focuses on ordering the radical numbers defined as ##x=\sqrt[7]{13}+\sqrt[6]{13}##, ##y=\sqrt[5]{13}+\sqrt[8]{13}##, and ##z=\sqrt[3]{13}+\sqrt[10]{13}##. Participants concluded that the function ##f(t) = 13^{1/t} + 13^{1/(13-t)}## is symmetric about ##t=6.5## and strictly monotonic in the intervals ##(0, 6.5)## and ##(6.5, 13)##. This leads to the ordering ##z > y > x## based on the behavior of the function and its derivatives. The discussion emphasizes the importance of understanding the concavity and monotonicity of the function to derive the correct order of the radical expressions.

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Apr 6, 2025

mathwonk

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how's this?
A = 1/5 - 1/7 > 1/6 - 1/8 = a, hence (13)^(1/5) - (13)^(1/7)

=[(13)^(1/7][13^A - 1] > [(13)^(1/8)][13^a - 1]

= (13)^(1/6) - (13)^(1/8), hence

y = (13)^(1/5) + (13)^(1/8) > (13)^(1/6) + (13)^(1/7) = x.

just elementary precalculus, unless I made a mistake, which I frequently do in this realm.

edit: I guess this is basically the same as pasmith's (more general) solution in the previous post.

Last edited: Apr 6, 2025