Powers of 10
- [Instructor] 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. So I could have just a 10. I could take two 10s and multiply them together, so 10 times 10, which you know is equal to 100. I could take three 10s and multiply them together. 10 times 10 times 10 is equal to 1000. And I could do this with any number of 10s. But at some point, if I'm doing this with enough 10s, it gets pretty hard to write. So for example, let's say I were to do this with 10 10s. So if I were to go 10 times 10 times 10 times 10, that's four, that's five, that's six, that's seven, that's eight, that's nine, that is 10 10s. Let's see, one, two, three, four, five, six, seven, eight, nine, 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 one followed by 10 zeroes. One, two, three, four, five, six, seven, eight, nine, 10. We put the commas there so it's just a little bit easier to read. This right over here is 10 billion, and it's already getting kinda hard to write, and imagine if we have 30 10s 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. So the way they do this is through something known as exponents. Exponents. And so 10 times 10, we can rewrite as being equal to, if I have two 10s 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 10s and multiply them together and we're going to get 100. Just so you're familiar with some of the parts of this, the two 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 1000? 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 10s and we see we're taking three 10s and we're multiplying them together. So this would be 10 to the third power. 10 is the base, three 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 1000. So this is really another way of writing 1000. 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 were taking 10 10s and multiplying them together. This is 10 to the 10th power. And 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 fifth 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 10s and multiply them together. So 10 times 10 times 10 times 10 times 10. And so 10 times 10 is 100, 100 times 10 is 1000, 1000 times 10 is 10000, 10000 times 10 is equal to 100000. So there you have it. You have the basics of exponents when we're dealing with 10, and I know what you were thinking. Can I put another number here instead of 10? And the simple answer is, you can, but we're not gonna cover that just yet in this video.