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### 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

[(-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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Exponents

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