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### The Nth Root of N

```
Date: 11/28/2000 at 20:25:09
From: Gavin King
Subject: Nth roots

Dear Dr. Math,

Is the nth root of n (a whole number other than 1) ever a rational
number?

Thanks, Gavin
```

```
Date: 11/28/2000 at 20:32:50
From: Doctor Schwa
Subject: Re: Nth roots

No. It definitely can't be rational. If you want a proof, try the same
proof that you have probably seen for the square root of 2 being
irrational.

If (a/b)^n = n, and a/b is in lowest terms, then a^n = n*b^n, so a is
divisible by n, ... or wait, it isn't, quite. For example, if n were 4
then a would only have to be divisible by 2 to make the left side
divisible by 4.

Okay, if p is a prime divisor of n, then p must be a prime divisor of
a also, whereupon you can prove that either n is divisible by p^n
(which it can't be, since p^n is too big) or b is divisible by p,
whereupon a/b is not in lowest terms after all, a contradiction.

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra

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