 So a very useful way of writing numbers involves what's known as exponents. So remember that arithmetic is bookkeeping. How many of what objects? And we use notation to help keep track of sets of similar things. So we might have a 1 and a 1 and a 1 and a 1 and a 1, but we could call this a 5. And mathematically we might write something like 1 plus 1 plus 1 plus 1 plus 1 equals 5. And this goes back to the basic idea that any whole number can be written as a sum of 1s. Or we might also look at a bunch of 5s. A 5 and a 5 and a 5 and a 5 is a... Well arithmetic is bookkeeping, this is 4 5s, and we could write this as 4 times 5. And we could write 5 plus 5 plus 5 plus 5 equals 4 times 5. And again, this goes back to the basic idea that any product of whole numbers can be written as a sum. While math ever generalizes, we always ask what happens if we try that with something new. So what if instead of a repeated sum, we had a repeated product? 5 times 5 times 5 times 5 times 5. Well obviously we can do the multiplication, but is there any way we can write this more simply? And to do that we introduce the notation for exponents. If I have a repeated product, I can write this in exponential notation. A to power n is the product of n a's. We say a, the thing that I'm multiplying together, is the base, and n, the number of times the factor appears, is the exponent or the power. And again it's useful to remember how you speak influences how you think. This thing should be read as a to power n or a to the nth power. For example, let's say I want to evaluate 5 to the second power. So definitions are the whole of mathematics, all else is commentary. So let's pull in our definition. So when we write something in exponential notation, the base is our repeated factor, and the exponent is a number of times that that factor appears. So in our expression 5 to the second, 5 is the base, so we're multiplying a bunch of 5's together, and 2 is the exponent, so we're multiplying 2 5's together. So 5 to the second is the product of 2 5's. And we can calculate that value. We should be able to go the other way and write something in exponential form. Again we see that we are multiplying a bunch of 2's together, so the base of the exponential expression will be 2, and arithmetic is bookkeeping, so we'll count the number of 2's that we have 1, 2, 3. And since there are 3 2's multiplied, the exponent will be 3. One of the common uses of exponents is to write a number in what's called prime factored form. No, that's not something you order online. What this goes back to is that we can always write a whole number as a product of primes. But there may be a lot of primes involved, so to keep the product from being too messy, we can write our product using exponents. So for example, let's find the prime factored form of 120, and to do that we begin by factoring 120. We want to write 120 as a product of prime numbers. So we might begin by writing 120 as a product of any numbers, about 12 times 10. Now if we can factor either of these numbers we should continue, so 12, well that's 3 times 4, and 10 is 2 times 5. Now if these factors 3, 2, and 5 can't be factored any further, but 4 can be, so we'll factor that as 2 times 2, and remember arithmetic is bookkeeping. We replace the 4 with the 2 times 2, but we had a 3, a 2, and a 5, and so we still have a 3, a 2, and a 5. And so now we take a look at our product, and remember we can write something in exponential notation whenever we have a repeated factor. And we only have 1, 3, and since arithmetic is bookkeeping we still have 1, 3. We only have 1, 5, and arithmetic is bookkeeping we still have 1, 5. But we have 1, 2, 3, 2s multiplied together. And so what this means is we can write this product of 2s as 2 to the third power, and that gives us our prime factorization. Now there's one last step, it's not really necessary, but it's traditional to order this in terms of the increasing size of the prime numbers. So this factor 2 to the third, we'll put that first, followed by 3, followed by 5.