Using exponents with powers of 10

So let's think about what happens when we multiply by 10, and see if we can see some type of a pattern. So let's just start with 1 and multiply that 1 by 10. And this, of course, is equal to 10. You knew that already. But let's multiply by 10 again. So let's do 1 times 10. And now let's multiply by another 10. Let's multiply by another 10. So what's this going to be? This is going to be 1 times 10, which is 10, and then that 10 times 10. So it's literally 10 10s or 100. Now let's multiply by 10 again. So we're going to have 1 times 10 times 10 times 10 times 10. What is this going to be equal to? Well we already know that 1 times 10 times 10 is 100, and then we're going to multiply that times 10 again. So we're going to literally have 10 100s which we know is the exact same thing as 1,000. So what's the pattern here? Well here we multiplied by 10 once and we have one zero. Here we multiplied by 10 twice and we have two zeroes. Here we multiplied by 10 three times and we have three zeroes in our answer. Let's do it again, although this looks like a pretty clear pattern. Let's multiply by 10 four times. So we'll start with a 1. We'll multiply by one 10, that's our first 10, then our second 10, then our third 10, and now our fourth 10. Well what's this going to be equal to? Well it's going to be this thing, I should say, times 10 or 1,000 times 10, which is 1,000s. So let me write that 10 1,000s. So once again, we started with a one, we multiplied by 10 four times, and we got 10,000. One followed by one, two, three, four zeros. Let me make that clear. We start with a 1. We multiplied by 10 one, two, three, four times, and our product is one followed by one, two, three, four zeros. So I think you see the pattern here. But there might be something you're craving. You're like, hey Sal, this is a pain to do this repeated multiplication. I wish there was some type of operation that just did it for us. The same way that back in the day when you were doing repeated addition where if you were saying 10 plus 10 plus 10 plus 10, you said, hey, this is the same thing as 4 times 10. There should be another thing for repeated multiplication. If I have 1 times 10 times 10 times 10 times 10, which is really the same thing as taking four 10s and multiplying them together because the 0 doesn't change the value. So 10 times 10 times 10 times 10. You're saying, hey, there should be some type of shorthand for this, and you're right. There is some type of shorthand for this and it's called an exponent. So the same way if you take four 10s and add them up, that's the same thing as 4 times 10. If you take four 10s and take their product, this is the same thing as 10 to the fourth power. 10 to the fourth power literally means taking four 10s and multiplying them together. Or you could view it as starting with a 1 and multiplying by 10 four times. So this is also equal to 10 to the fourth power. This right over here, we started with a 1. We multiplied by 10 three times, which is the same thing as taking three 10s and multiplying them together. This is 10 to the third power. This right over here is 10 squared, or 10 to the second power. And this right over here, we just have one 10 right over here. We just started with the one and we multiplied by 10 once. This is 10 to the first power. And just so you know a little bit of the terminology, here 10 is our base. The base is 10. And this 4 that we have right over here, this is the exponent. So given that notation, let's just do one practice. Let's think about what-- I'll do new colors-- what 10 to the sixth power is. And I encourage you to pause the video and think about it. Well there's two ways to think about it. You could view this as six 10s-- so 10, 10, 10, 10, 10, and 10 and multiplying them together. So we multiply them all together. Or you could view it as starting with a 1 and multiplying it by 10 six times. But either way, what are you going to get? Well if you're multiplying 1 by 10 six times, you're going to end up with 1 followed by six zeroes. We already saw this pattern. So it's going to be 1 followed by one, two, three, four, five, six zeros, or one million. So 10 to the sixth power is the same thing as 1 being multiplied by 10 six times. We're taking six 10s and multiplying them together. And that's going to be equal to 1 followed by six zeros, which is this pattern we saw or one million.