Associated Topics || Dr. Math Home || Search Dr. Math

### Irrational Powers

```
Date: 8/30/96 at 22:20:49
From: Anonymous
Subject: Irrational numbers to irrational powers

Does an irrational number to the irrational power yield a rational
number?  If so, what are the numbers?  The two irrational numbers
don't need to be the same.

I approached the problem through trial and error, but it is too time
consuming.
```

```
Date: 9/5/96 at 10:13:58
From: Doctor Jerry
Subject: Re: Irrational numbers to irrational powers

In general, an irrational number to an irrational power yields an
irrational number.  The mathematics needed to prove this is quite
complicated.  I'll try to explain a little.

A mathematician named Ferdinand Lindemann proved that if
x1, x2, ..., xn are distinct algebraic numbers (see below) and
p1, p2, ...,pn are algebraic numbers, not all zero, then the sum

p1*e^(x1) + p2*e^(x2) + ... + pn*e^(xn)

cannot be zero.

A number is an algebraic number if it is a root of an equation

a0 + a1*x^1 + a2*x^2 + ... +an*x^n = 0,

where a0, a2, a2, ... , an are rational numbers.  All of the rational
numbers and many of the irrational numbers are algebraic.

For example, the number 2^(1/2) is algebraic since it is a root of
x^2 - 2 = 0.

If we take n = 2, p1 = 1, and x2 = 0, then e^(x1)+ p2 cannot be zero
if x1 is a nonzero algebraic number and p1 is any algebraic number.
From this we conclude that e^(x1) cannot be rational, and so must be
irrational.  The number e is irrational and we may take x1 = 2^(1/2).
Hence, an irrational number to an irrational power need not be
rational.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 9/5/96 at 10:17:12
From: Doctor Ken
Subject: Re: algebra(rational and irrational numbers)

Hello -

If you're looking for some specific examples of irrational numbers
x and y such that x^y is rational, you can probably construct some
yourself.  Here's how you might go about doing it: take your favorite
irrational number and call it x.  Now ask yourself what number y would
have to be in order to satisfy x^y = 5 (Think logs!).

Good luck!

-Doctor Ken,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Modern Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search