Simplifying this absolute value

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Discussion Overview

The discussion revolves around the manipulation of the expression ##|nr^n|^{1/n}##, specifically whether the exponent can be moved inside the absolute value when dealing with a rational exponent. Participants explore the implications of this manipulation under different conditions for the variables involved.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if ##|nr^n|^{1/n} = |(nr^n)^{1/n}|## is valid for rational exponents.
  • Another participant asserts that this manipulation holds in the real numbers.
  • A subsequent reply suggests testing specific values, such as ##r = 1## and ##n = -2##, to investigate the validity of the claim.
  • There is a suggestion that the equality may not hold generally, prompting further inquiry about the case when ##n## is positive.
  • One participant states that if both ##n## and ##r## are positive, the absolute value can be ignored.
  • Another participant clarifies that while ##n## is positive, ##r## is not necessarily positive, which may affect the validity of ignoring the absolute value.

Areas of Agreement / Disagreement

Participants express differing views on whether the exponent can be moved inside the absolute value, particularly under varying conditions for the variables. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants have not fully explored the implications of negative values for ##r## or the specific conditions under which the manipulation holds, leaving some assumptions unaddressed.

Mr Davis 97
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I have the expression ##|nr^n|^{1/n}##. A quick question is whether I can allow the exponent to go inside of the absolute value. I know that if it were an positive integral exponent then because of the multiplicativity of the absolute value function that would be allowed. But I'm not sure what I'm allowed to do in the case of this rational exponent...
 
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In the real numbers that works.
 
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Mr Davis 97 said:
whether I can allow the exponent to go inside of the absolute value.

Are you asking whether ##|n r^n|^{1/n} = | (n r^n)^{1/n}| ## ?

Try ##r = 1,\ n = -2##.
 
Stephen Tashi said:
Are you asking whether ##|n r^n|^{1/n} = | (n r^n)^{1/n}| ## ?

Try ##r = 1,\ n = -2##.
So it's not generally true then? What if ##n## is positive?
 
If n and r are both positive then absolute value can be ignored.
 
mathman said:
If n and r are both positive then absolute value can be ignored.
Well ##n## is positive while ##r## is not necessarily positive.
 

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