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/