Zero Exponent and Negative Exponents

Just a short post to follow up on my last one about the properties of exponents.  In that post, I explained to you various properties of exponents that allow you to do arithmetic with them to simplify them into easier to work with expressions.  However, I only used positive exponents in my explanations.  Here, I need to briefly explain two more concepts: zero exponents and negative exponents.

Zero exponents are simple.  They follow one single rule that applies wherever you will see them.  No matter what the base is, if it is as simple as x, or as complicated as (45xyz / (412ab+26c)), if you raise the base to a power of zero, the expression equals 1.

a0 = 1

It is a very handy shortcut, so whenever you see it, you can make life a lot easier for yourself if you just simplify the expression to 1.  For example, you don't want to get stuck rearranging to solve for a in that 45xyz... example above, when you can simplify the whole thing to 1.

So that is zero exponents.

Next is negative exponents.  They aren't QUITE as simple, but the trick to working with them is.  And here it is:

Whenever you see a negative exponent, all you do is make it positive, but put the whole thing on the bottom of a fraction under 1, like so:

a-n = 1 / an

Once it's like that, you can apply any or all of the properties of exponents to it without having to deal with the negative anymore.

Just to put some numbers to it, here is an example for you to see what I mean:

(a2b3)-4
= 1 / (a2b3)4
= 1 / a8b12

Hopefully you can see what I mean with these examples.  They are just a few shortcuts that will make working with exponents a lot easier for you.  Let me know if you need any more explanation about these topics, but for now I will leave it at that.

1 comment:

1. Perhaps you may wish to add that for a^n where a is positive and n is any real numbers, a^n will always be greater than zero due to the asymptotic nature of exponential curves when near the x-axis.