How do indices work?

Starting with multiplication how can we rewrite (click to see):

Click on the equation or click here for the answer.

Did you know the answer? The rule is that when you mutliply two powers, and the bases are the same, then you add the indices. If you knew this then good. Now do you know why that is the thing to do?

Why we need to know why.

Here is a description of why you add the indices

Notice that in the first and third lines we used the definition of indices. In the second line we used the fact that multiplications can be done in any order whatsoever. Once you know this the "add the indices" rule follows from the fact that addition is a short cut way to count up the factors.

Explain these reasons.

We need to know that multiplication can be done in any order because the order of operation changes when we add the indices. With we only ever multiply by 2, since firstly we do , then , then and so on. With one of the multiplications is . Check out the following expression to see the different order of operations. Click on a number other than 2 to factorize. Click on an operation to implement that operation and all the others that precede it. To get the answer 128 you must click on the operation that is done last. Which one is the last?

Here is a formal statement of the rule for adding indices.

Add the Indices Rule (Formal Statement).