### and negative powers.

In college I was told that , and I remember why that is the case being explained. However, the finer details escaped me and I quickly forgot 'why' or how we can show this. Today, I once again came across this fact and it is them neat little mathematical things, that make you go 'awe'. ;) [Well it came across as cool!]

We have to know about certain rules before proceeding, but I will only be stating the few used.

(a) Also, *a* must not equal to zero. [Cancellation law for multiplication].

(b)

(c)

Now there are a few ways to go about doing this, but it depends on what you already take for granted I suppose. I will do it in the way I read it from the book first, because that is needed for negative powers.

From (b), . Hence, (a) [the multiplication cancellation laws] then tell us that no matter what a is, .

Now for the negative powers, we use the 'infamous trick' of adding in zero. [We actually used it in a lecture today - bringing a wry smile to my face.]

We start from knowing what a^k is equal to, then: . (By using the rules mentioned above). So from this we can see that .

Neat.

If we know about negative powers, we can always use that fact to show that .

Consider: .

It does seem circular, but I guess you don't have to be 'that' rigorous. It's quite embarrassing, the way I had the 'ah ha' moment when I recalled the above. We take a lot of things for granted in maths, and 'nice' little things like the above are just 'nice'.

## 4 comments:

I feel 'naughty' because I started the post title with a^0=1!!

"a" should be assumed nonzero

(check the "cancellation" law

in particular). yrs in the faith.

v.

In your condition (a) you need to require that

a is not equal to zero.

I have corrected it now. Many thanks to both of you.

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