Restrictions on Roots
Date: 06/20/2002 at 01:28:49 From: Todd Subject: Definition of "root" When we talk about the nth root of a number, are there any restrictions on n? (I realize the trivial case that n cannot be zero.) I tried searching the Internet using www.google.com, and I tried searching the Dr.Math FAQ, but I must not be asking the question correctly. For example, since 5^(1000/827) is close to seven, can we say that the [827/1000]th root of seven is close to five? Or is it nonsensical to talk about non-whole-numbered roots of numbers? - Todd
Date: 06/20/2002 at 08:43:59 From: Doctor Peterson Subject: Re: Definition of "root" Hi, Todd. I wouldn't say this is nonsensical; anyone could understand what you meant if you said it. But it's not used very much. However, I don't think you said quite what you meant; you should have said the 1000/827 root of 7 is about 5, since the 1000/827 power of 5 is about 7. That is, the nth root of x can be defined as the 1/nth power of x, or alternatively as the solution of the equation x = y^n. To give another example, since 9^(3/2) = 27 we could say that 9 is the 3/2 root (not the 2/3 root!) of 27. That would simply mean that 9 is the number whose 3/2 power is 27. We could also say that the 2/3 root of 9 is 27, since 9^(1/(2/3)) is 27. But we usually avoid this, since on one hand it is not necessary (roots being just a shorthand for reciprocal powers), and on the other hand it is potentially confusing (as we've just seen!). It's better to stick to exponential notation, apart from familiar cases like square roots. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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