Introduction to powers of 10

In this video, I'm going to introduce you to a new type of mathematical notation that will seem fancy at first, but hopefully, you'll appreciate is pretty useful and also pretty straightforward. So let's just start with some things that we already know.

I could have just a ten. I could take two tens and multiply them together. So, ten times ten, which you know is equal to a hundred. I could take three tens and multiply them together. Ten times ten times ten is equal to 1000. I could do this with any number of tens, but at some point, if I'm doing this with enough tens, it gets pretty hard to write.

For example, let's say I were to do this with ten tens. So, if I were to do ten times ten times ten times ten, that's four, that's five, that's six, that's seven, that's eight, that's nine, that is ten tens. Let's see: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. This is going to be equal to, even the number that it's equal to is going to be quite hard to write. It's going to be 1 followed by 10 zeros: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Let me put the commas there so it's a little bit easier to read. This right over here is 10 billion, and it's already getting kind of hard to write. Imagine if we had 30 tens that we were multiplying together.

So, mathematicians have come up with a notation and some ideas to be able to write things like this a little bit more elegantly. The way they do this is through something known as exponents. Exponents. So, 10 times 10, we can rewrite as being equal to. If I have two tens and I'm multiplying them together, I could write this as 10 to the second power. That's how someone would say it; they would say 10 to the second power. That looks fancy, but all that means is let's take two tens and multiply them together, and we're going to get 100.

Just so you're familiar with some of the parts of this, the 2 would be called the exponent and the 10 would be the base. So, 10 to the second power is 10 times 10 is equal to 100.

So how would you write 10 times 10 times 10 or a thousand? How would you write that using exponents? Pause this video and see if you can figure that out.

Well, as exactly as you might have imagined, we're taking a certain number of tens, and we see we're taking three tens and we're multiplying them together. So, this would be 10 to the third power. 10 is the base; 3 is the exponent. We would read this as 10 to the third power. If you ever saw 10 to the third power, that means, "Hey, let me multiply 10 times 10 times 10." That's the same thing as a thousand. So this is really another way of writing a thousand.

And what about this number here, 10 billion? What's a way that we could write it using exponents? Pause the video and figure that out.

Well, as you might have imagined, we're taking 10 tens and multiplying them together. This is 10 to the 10th power. We can go the other way around. If someone were to walk up to you on the street and say, "What is 10 to the 5th power?" What is that? What number that you're probably familiar with would this be?

Well, this would mean that we're going to take five tens and multiply them together. So, 10 times 10 times 10 times 10 times 10. And so, 10 times 10 is 100. A hundred times ten is a thousand. A thousand times ten is ten thousand. Ten thousand times ten is equal to one hundred thousand.

So there you have it. You have the basics of exponents when we're dealing with 10. And I know you were thinking, "Can I put another number here instead of 10?" And the simple answer is you can, but we're not going to cover that just yet in this video.