I'll discuss some rules for working with * integer
exponents*. Actually, these rules work with arbitrary exponents,
but it is easier to explain why they're true in this case. So in what
follows, all the powers are assumed to be positive or negative
integers.

Let's do a specific example to check that this makes sense.

*Negative exponents translate to reciprocals.* That is,

So

Here is a rule for raising a power to a power:

Again, I'll do a specific example to check that this makes sense.

Now I'll use these rules to simplify some expressions.

* Example.*

Warning: . The expression " " is read to mean that you square the 3
*first*, then negate the result. On the other hand, .

A number raised to an even power is always positive:

On the other hand, a *negative* number raised to an odd power
is negative:

* Example.*

(Whether you write or depends on what you're using the expression for.)

By the way, these rules work with non-integer exponents. For example,

Here's a specific case of the first rule so you can see that it makes sense.

* Example.* Simplify and write the result using
positive powers.

You can derive a rule for division from the rule for multiplication:

* Example.*

But

The things being raised to powers (a and b) are not the same, so the rules I've given don't apply.

* Example.* Simplify, writing the result in
terms of positive powers:

* Example.* Simplify, writing the result in
terms of positive powers:

* Example.* Simplify, writing the result in
terms of positive powers:

* Example.*

* Example.* Simplify, writing the result in
terms of positive powers:

* Example.* Simplify, writing the result in
terms of positive powers:

Copyright 2008 by Bruce Ikenaga