# Transcendental numbers and power terms

• January 15th, 2008, 01:05 PM
sunshinewarrior
Transcendental numbers and power terms
Hi all

Perhaps a silly question/speculation but I wondered if any of you might be able to set me right.

Consider the transcendental number 2<sup>√2</sup>.

Now

√2 = 2<sup>0.5</sup>

and

0.5 = ½ = 2<sup>-1</sup>

So is

2<sup>√2</sup> = 2<sup>2<sup>2<sup>-1</sup></sup></sup> ?

It seems to work but doesn’t it just seem a bit dodgy too? And if it is correct, does that mean that power terms always have to be evaluated from the top down?

Help!

cheer

shanks
• January 15th, 2008, 01:25 PM
serpicojr
Indeed those are both valid expressions for the same number. And you're correct that exponents have to be evaluated "top down". To evaluate something like:

2<sup>expression</sup>

you first have to calculate the expression. In this case, your expression is another exponent, so you have to calculate that one first... and to do that, you have to calculate another exponential expression.
• January 15th, 2008, 01:59 PM
sunshinewarrior
So in what circumstances do you multiply powers?

That is, what sort of equation might lead to the term:

x<sup>n.m</sup>?
• January 15th, 2008, 03:00 PM
serpicojr
The rule for multiplying powers is:

x<sup>yz</sup> = (x<sup>y</sup>)<sup>z</sup>

This is usually phrased as "raising a power to a power", although this can be a little confusing, as is the first instance of "power" refers to the expression x<sup>y</sup> and the second instance refers to the exponent z.

This rule and the addition rule:

x<sup>y+z</sup> = x<sup>y</sup>x<sup>z</sup>

are easy to show for integers y and z, and the facts for general real numbers basically follow from definitions and the facts for integers.
• January 16th, 2008, 05:39 AM
sunshinewarrior
Quote:

Originally Posted by serpicojr
The rule for multiplying powers is:

x<sup>yz</sup> = (x<sup>y</sup>)<sup>z</sup>

This is usually phrased as "raising a power to a power", although this can be a little confusing, as is the first instance of "power" refers to the expression x<sup>y</sup> and the second instance refers to the exponent z.

That's what I thought. Because here it seems as though it's a licence to collapse my original expression to something absurd but, as you point out, that's not allowed.

Thanks for the time and explanations.

cheer

shanks