Exponents and powers of 10 patterns

We are asked what is 10 to the fifth power equivalent to. Well, 10 to the fifth power is the same thing as taking a 1 and multiplying it by 10 five times. So let's do that. So that's three, four, and five. So it's 1 times 10 times 10 times 10 times 10 times 10. Notice one, two, three, four, five. And what's this going to be? Well, 1 times 10 is 10. 10 times 10 is 100. 100 times 10 is 1,000. 1,000 times 10 is 10,000. 10,000 times 10 is 100,000. So this is going to be 100,000. Now, you might have noticed, every time we multiply it by 10, we're adding another 0 to the product. So if we're multiplying by 10 five times, we're going to add five 0's to the product. So this is literally going to be 1 followed by five 0's, one, two, three, four, five. So 10 to the fifth is the same thing as 100,000. Let's do another one that is on a similar topic. How many 0's does the product 67 times 10 to the fifth have? Well, there's a bunch of ways of thinking about that. 67 times 10 to the fifth is the exact same thing, this is equivalent to-- you could view this as 67 times. And we could use the same exact logic that we just saw in the previous problem. So 67 times 1 times 10 times 10 times 10 times 10 times 10. And we already figured out exactly what this is. If you have 1 multiplied by 10 five times, this right over here is going to be 1 followed by five 0's. One, two, three, four, five. Or it's going to be 100,000. So there's a couple ways to think about it. If you're looking at just this product right over here, you could say, well, 67 times 1 is just going to be 67. And then every time you multiply by 10, you're going to add another 0. So you could say, well, this is just going to be 67. And we're going to be multiplying by 10 five times, so we're going to add five 0's here. One, two, three, four, five. And so this would come to 6,700,000. Another way you could think about it-- this is the same thing as 67 times 100,000, so it's going to be 67, and then we have five 0's here. So once again, one, two, three, four, five. You get the exact same value. Let's do another one. How many 0's does the quotient 5,700,000 divided by 10 to the third power have? Well, 10 to the third power we already know. 10 to the third power-- that's the same thing as 1 times 10 times 10 times 10, which is the same thing as 1,000. And so if we're dividing by that, that's the same thing. So another way of writing this expression right over here. So 5,700,000 divided by 10 to the third is the same thing as-- I could write this way. 5,700,000 divided by 10 times 10 times 10. We could put a 1 out here, but this won't really change the value, which is equivalent to 5,700,000 divided by 1,000. And either way you think about it, every time you divide by 10, you're going to eliminate one of these 0's. So if you divide by 10 three times-- so if you divide by 10 once, you're going to eliminate a 0. Divide by 10 again, you're going to eliminate another 0. Divide by 10 again, you're going to eliminate another 0. You're going to be left with 5,700. Another way of thinking about it is, well, if I'm dividing by something that has three 0's, I'm going to eliminate three 0's. So if I eliminate three 0's, I am left with 5,700. Now, just to be clear, I could only do this because this was 1,000. If this was like 3,000 or something, I could think of this as 3 times 1,000. And then I could only cancel out the 1,000 with these 0's. But the 3 I would then have to divide separately. But only because I'm straight up dividing by 1,000. I'm dividing by a power of 10. I have three 0's here. I can cancel it out with three 0's there. Let's do another one. So it asks you to do a few. And these are the type of questions that you might see in the exercise. When 72.1 is multiplied by 10 to the third, the decimal point moves blank places to the blank. And in the exercise, you might see a dropdown here. So remember, if you're multiplying by-- and this is essentially 10 to the third, so you're multiplying by 1,000, you're going to get a larger number. You're not going to get a smaller number. And so every time you multiply by 10, your decimal point is going to move to the right because you're getting larger. So we're going to move the decimal point. So 72.1, if we multiplied it by 10 once, then we would move the decimal point over once. And we'd get 721, which makes sense. 72 times 10 is 720. 72.1 times 10 should be 721. But if we want to multiply it by 10 three times, we're going to move the decimal place over not just once, but twice, and three times. And you say wait, wait. How could I move it over? There's nothing here. Well, you throw a 0 in there. And so you're going to get, it's going to be equal to-- so it's going to be equal to 72,100. Now, they're not asking us that. They're just asking how we would do it. And we saw the decimal point moves three places to the right. And the big key here is you're going to get a larger number when you multiply by 10 three times, or multiply by 1,000. Let's do this one. When 56 is divided by 10 to the third, the decimal point moves blank places to the blank. Well, dividing by 10 to the third is the same thing as dividing by 10 three times. And every time you divide by 10, it's going to become a smaller number. So 56, and you might say, where is the decimal point? Well, there's implicitly a decimal point right over there. You divide by 10 once, you're going to get a smaller number. 56 is going to become 5 and something, so it's going to become literally 5.6. Divide by 10 again, it's going to become 0.56. Divide by 10 again, you're like, wait. If I keep moving the decimal place again to the left, what am I moving it to the left of? Well, you could throw a 0 right over here. And so if you move the decimal point three places to the left, you are left with 0.056. And you might want to put a 0 out here just for clarity. But what we did here is we moved the decimal point three places to the left in this situation. We got a smaller value.