Multiplying and dividing by powers of 10

In another video, we introduce ourselves to the idea of powers of 10. We saw that if I were to just say 10 to the first power, that means that we're just really just going to take 1.

If we have 10 to the second power, that means that we're going to take two tens, so a 10 and a 10, and we're going to multiply them. So that's going to be 100. If we take 10 to the third power, that's going to be three tens multiplied together, which is equal to one thousand, which is also one followed by three zeros.

So you're already starting to see a pattern. What we're going to do in this video is think about patterns when we multiply arbitrary things or divide our very things by powers of ten.

So let's start with a number. Let's say I will start with 2.3. And let's first just multiply it by 10 to the first power. Well, that's the same thing as just multiplying it by 10. We've seen already when you multiply by 10, you shift all the digits one place to the left.

So the two, which is in the ones place, will end up in the tens place, and then three, which is in the tenths place, will end up in the ones place. So this is just going to be equal to 23. And it's always good to do a little bit of a reality check. If I just had two and if I were to multiply it by 10, you'd say, "Okay, that's about 20." So it makes sense that 2.3 times 10 is 23.

But let's keep going. Now let's multiply 2.3 not by 10 to the first power, which is just 10, but let's multiply it times 10 to the second power. What is that going to be? Pause this video and see if you can figure that out.

All right, well 10 to the second power, we already know that's equal to 100. And so when you multiply by a hundred, or you multiply by ten twice, you're just going to shift every digit two places to the left. So let me draw some places here.

So the thing that is in the ones place will go to the hundreds place, and the thing that is in the tenths place will go to the tens place. And so this two will now be two hundreds. This three tenths will now be three tens, and we now have zero ones.

And then if we were to multiply by ten again, so if we were to say 2.3 times 10 to the third power, well then we're going to shift everything three places to the left. 10 to the third power, this is the same thing as multiplying by one thousand.

So 2.3 times 10 to the third, the two is going to be shifted three places to the left. So the two is going to become two thousand, which makes sense. So it's going to be two thousand. The three tenths is going to shift two places to the left, so it's going to be three hundred, and then we now have zero tens and zero ones.

So the pattern that you've probably seen is if you multiply a number times 10 to some power, you are just shifting the digits to the left by that power.

And if we divide by a power of 10, the same thing would happen, but we would now be shifting our digits places to the right. So for example, what is 2.3 divided by 10 to the first power? Pause this video and try to figure that out.

Well, 10 to the first power is the same thing as 10. So when we divide by 10, all of our digits are just going to shift one place to the right. So this two is going to end up in the tenths place, and the three is going to end up in the hundredths place. So this is going to give us 0.23.

Two is now in the tenths; three is now in the hundredths. But we could keep going. What if we were to say 2.3 divided by 10 to the second power? Pause this video, try to figure out what that is.

Well, in this situation, we are going to shift all the digits two places to the right. So let me put my places here. So that's the ones, tenths, hundredths, thousandths.

And so the thing that's in the ones place is two; it won't just go to the tenths. It'll go to the hundredths place, which you don't quite see here; it's right over there. So the 2 is going to show up here, and then the 3 is going to shift 2 places to the right, so it's going to end up there, and we have 0 ones and we have 0 tens.

So you're probably already seeing the pattern here. Again, whatever the exponent is, if you're dividing by 10 to that power, you're going to shift that power, that exponent, that many times. You're going to shift the digit that many places to the right.