Raising a Power to a Power

Date: 11/27/97 at 14:21:57
From: Lucas Overmire
Subject: (Algebra) raising a power to a power

I know that this answer is right but I just can't seem to figure out 
why.  Could you please help me?

[(-x)to the 5 power]to the 8 power. My teacher said that the answer 
was x to the 40 power but I don't understand why it isn't -x.

Thanks for any help.
Lucas Overmire

Date: 11/27/97 at 18:21:49
From: Doctor Charles
Subject: Re: (Algebra) raising a power to a power

(-x) ^ 5 = (-x) * (-x) * (-x) * (-x) * (-x)

((-x)^5)^8 = ((-x)^5) * ((-x)^5) * ((-x)^5) * ((-x)^5) * 
             ((-x)^5) * ((-x)^5) * ((-x)^5) * ((-x)^5)

           = (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x) *
             (-x) * (-x) * (-x) * (-x) * (-x)

           = (-x) ^ 40             
           
 or        = (-x * -x) ^ 20
           = ( x ^ 2 ) ^ 20
           = x ^ 40

If we ended up with an odd power things would have been different:
for example ((-x) ^ 5) ^ 7 = (-x) ^ 35
                           = (-x) * ((-x) ^ 34)
                           = (-x) * (-x * -x) ^ 17
                           = (-x) * ( x ^ 2) ^ 17
                           = (-x) * x^34
                           = - (x * x^34)
                           = - x^35

In general (-x)^n =  x^n if n is even and:
           (-x)^n = -x^n if n is odd.

We can prove this. If n is odd then n = 2*m+1 for some m
       (-x)^n = (-x)^(2m+1)
              = (-x) * (-x) ^ 2m
              = (-x) * ((-x)^2)^m
              = (-x) * (x^2)^m
              = (-x) * x^2m
              = - x^(2m+1)
              = - x^n
If n is even then n = 2m for some m so:
       (-x)^n = (-x) ^ 2m
              = ((-x)^2)^m
              = (x^2)^m
              = x^2m
              = x^n

-Doctor Charles,  The Math Forum
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