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

### Are Exponents Associative?

```
Date: 02/05/2002 at 07:43:41
From: Lucas
Subject: Complex numbers

I have a question that I couldn't answer:

How much is 2^(i) or X^(i) ?

I asked my teacher, and he said that

2^(i) = 2^(-1)^(0.5)
= 0.5^0.5~0,707106781186547524400844362104849

I think a formula exists to find y and z:

2^(i) = y and y^(i) = 0.5
2^(-i) = z and y^(-i) = 0.5
2^(-i) = z and y^(i) = 2

Is this correct? If not, why not?
Thanks.
```

```
Date: 02/05/2002 at 11:37:37
From: Doctor Peterson
Subject: Re: Complex numbers

Hi, Lucas.

It looks as if your teacher accidentally assumed that exponents are
associative, that is, that

(a^b)^c = a^(b^c)

We are used to this being true for addition and multiplication, but it
is NOT TRUE of exponents; for that reason, it is important to include
parentheses to clarify this, though by convention we take a^b^c to
mean a^(b^c). Note that, in reality,

(a^b)^c = a^(b*c)

(Your own facts are correct, if I fix one typo, and they make use of
this fact. Unfortunately, I don't see that they go anywhere useful.)

Here is what your teacher claimed, with the wrong move stated
explicitly:

2^i = 2^[(-1)^0.5] = [2^-1]^0.5 = 0.5^0.5 = 0.707
NO!

To solve this correctly, you need to use Euler's formula, which you

Imaginary Exponents and Euler's Equation
http://mathforum.org/dr.math/faq/faq.euler.equation.html

This says that

e^(xi) = cos(x) + sin(x) i

We can transform your power of 2 to a power of e this way:

2^i = (e^ln(2))^i = e^(ln(2) i)

Now apply Euler to this:

= cos(ln(2)) + sin(ln(2)) i

= 0.769 + 0.639 i

I know that sounds weird, but this definition of imaginary powers
makes all the rules for exponents work, so it is what we use. You can

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Imaginary/Complex Numbers

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