# The basics of Exponents

(source: http://www.purplemath.com/modules/exponent.htm)

# Some Classic Examples

Try this one:

So, can you make a rule about multiplying exponents? Write it down, in your own words. Something like, "when you multiply exponents with like bases, you...." Got it? Exactly.

Here's another type of question:

(source: http://www.purplemath.com/modules/exponent.htm)

Here we have raised an exponent to some other exponent. What, in practical terms, have we done with the two exponents?

# Some Tricks to Watch out for

Let's try this example:

Two things to think about here: what is the rule for raising an exponent to some other exponent? What do you need to watch our for? Hint: what happens to the 7?

### WARNING!

What happens in this example:

(3 + 4)² = ?

I can confidently predict that you will get one of two answers: either 25 or 49. So: which is correct? In the last example, we were able to distribute the exponent inside the parentheses to get the solution 49x4y2. The question is: are we allowed to distribute an exponent inside parantheses when we have an addition sign?

The answer, of course, is no.

(3 + 4)² is actually equal to (3 + 4)(3 + 4), or (7)(7) which is also 7², and if we evaluate that properly, the answer will be 49.

Another trick you'll see is this one:

Is there any difference between these?

-3² and (-3)²

Is the answer for each the same?

Take some time and think it through...

The answer is, no, they are not the same.

In the first example, we need to do the exponent first. So we take 3 and square it, and get 9. But we still have that negative sign. The negative is applied through multiplication as such:

(-1)(9) = - 9

In the other example, we have the negative sign inside the brackets, and so it gets run through with the exponent as such:

(-3)(-3) = 9

Clear as mud, right? Right. It's a tricky thing, that negative sign. The bottom line is, unless you have a negative sign inside the brackets, you have to think of it as such:

-3² = -(3²), which gives us -9.

Onward!