Order the given radical numbers

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Homework Help Overview

The discussion revolves around ordering three radical expressions: ##x=\sqrt[7]{13}+\sqrt[6]{13}##, ##y=\sqrt[5]{13}+\sqrt[8]{13}##, and ##z=\sqrt[3]{13}+\sqrt[10]{13}##. Participants explore various mathematical approaches to compare these expressions without using a calculator.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants consider using arithmetic and geometric means but are advised against it. The idea of comparing the sums of the orders of the radicals is introduced. Some participants discuss the behavior of a function related to the expressions and its concavity. Others question the implications of symmetry and the behavior of the function in different intervals.

Discussion Status

The discussion is ongoing, with various methods being explored, including the use of derivatives and the properties of the function defined as ##f(t) = 13^{1/t} + 13^{1/(13-t)}##. Some participants express uncertainty about the implications of their findings, while others provide insights into the function's behavior.

Contextual Notes

Participants note that all methods are allowed, and there is a focus on the interval ##(0, 13)## where the function is well-behaved. The discussion acknowledges the presence of poles at ##x=0## and ##x=13## that should be avoided.

  • #31
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.
 
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