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Thread: Transcendental numbers and power terms

  1. #1 Transcendental numbers and power terms 
    Forum Professor sunshinewarrior's Avatar
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    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


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  3. #2  
    Forum Professor serpicojr's Avatar
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    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.


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  4. #3  
    Forum Professor sunshinewarrior's Avatar
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    So in what circumstances do you multiply powers?

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

    x<sup>n.m</sup>?
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  5. #4  
    Forum Professor serpicojr's Avatar
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    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.
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  6. #5  
    Forum Professor sunshinewarrior's Avatar
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    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
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