How to Use the Index Law of Addition

Now, a cubed is a short way of saying A times A times A, but what if we were asked to simplify A to the power of 3 times A to the power of 4? I’ll do my working out for this the long way. See if you can see any short cuts. We’ve just said that A to the power of 3 means A times A times A, and it has to be multiplied by A to the power of 4, that is A times A times A times A. Can you see how to write this expression down in a simpler form? Well, if we count the A’s, we get A to the power of 7. Without multiplying all the A’s out, how can we do this problem more quickly? Did you notice that if we add the indices 3 and 4, we get 7? So A to the power of 3 times A to the power of 4 equals A to the power of 3 plus 4 which equals A to the power of 7. Now, mathematicians have found that this shortcut is one of the laws of math. It’s the first of what we call the index laws. A cubed and A to the fourth had the same base, A. So when we multiply them together, we were able to just add the two indices. From this, we can say our first index law is when multiplying terms that have the same base, we add the indices. This can be written as a general rule. A to the power of N times A to the power of N equals A to the power of N plus N.