Hi, this is Anthony Varela, and today we're going to multiply and divide numbers written in scientific notation. So we're going to review scientific notation, then we're going to practice multiplying and dividing in scientific notation, and we're going to be applying our properties of exponents as we do this. So first, let's review scientific notation. Scientific notation puts numbers in this form. We have a times 10 raised to the power of b. So positive powers of 10, that would be when b is a positive number, we're talking about large numbers. And negative powers of 10, so if b is a negative number, we're talking about small numbers.
Now, a can only have one single digit to the left of the decimal. There can be as many as you want to the right of the decimal, but we're only allowed one digit to the left of the decimal. And that single digit can not be zero, so it has to be 1 through 9. So let's take a look at this number 132 times 10 to the 4th. This is not written in proper scientific notation, because we have more than one digit to the left of this invisible decimal right here. So we have to move that decimal over, and we have to move it over-- let's see-- 1, 2 places so we have a single non-zero digit here to the left. And because we moved our decimal, we have to adjust that exponent. We have to change it from a 4 to a 6. Now, that's because when we're moving our decimal to the left, we increase our exponent. We move it to the left 2 places, so exponent went up by 2.
Another example of a number not written in proper scientific notation, we have a single digit here, but it's 0, and remember it can't be 0. So we have to move this decimal over to the right 1 place. So now that we have 8.7, but we have to adjust our power of 10. When we're moving it to the right, we have to decrease our exponent. Since we moved to the right 1 place, we bring our exponent down by 1.
All right, so now let's practice multiplying 2 numbers that are written in scientific notation. So here we have 1.32 times 10 to the 6th. We want to multiply that by 8.7 times 10 to the 2nd. So because multiplication is commutative, I can multiply these numbers in any order I want. So what I'm going to do is group then my decimal numbers together, and then I'm going to place my powers of 10 together. So I can multiply the decimals, then I'm going to multiply the powers of 10.
So in multiplication, multiply the decimals then the powers of 10. So when you multiply 1.32 by 8.7, we get 11.484, and now we're going to multiplayer powers of 10. We're going to apply a property of exponents. So we can do this in scientific notation because our base is always 10, always the same base. And we're going to use the product of powers, which tells us that if our bases are the same, we can just add those powers. So we have 11.484 times 10 to the 8th. 6 plus 2 is 8. But notice 11.484 is not in proper scientific notation. Let's move our decimal and adjust our exponent. Move to the left to increase our exponent.
Let's divide numbers in scientific notation. So 3.7 times 10 to the 5th divided by 1.66 times 10 to the 2nd. And there's a same idea here with multiplication. When we're dividing 2 numbers, we're going to divide our decimals, then we're going to divide our powers of 10. And we connect them then with multiplication, because remember scientific notation has a decimal a value multiplied by a power of 10. So now we just need to then divide the decimals, and then we'll divide the powers of 10.
Well, 3.7 divided by 1.66 is 2.23 when we round, and now we're going to divide our powers of 10. Well, because in scientific notation our bases are always 10, we have common bases, we can use the quotient of powers property, which says if we're dividing 2 exponentials with the same base, we can take the 1st exponent and subtract then the 2nd exponent. So 5 minus 2 equals 3. And this is written in proper scientific notation, a single non-zero digit to the left of the decimal. No adjustment needed.
We're going to go through a final example of squaring numbers in scientific notation. And what you could do is you can write this out as a multiplication problem and then use the product of powers property, but I'm going to show you something different. When we're squaring numbers in scientific notation, we're going to distribute that outside exponent of 2 into both the decimal number and our power of 10. So when we're squaring in scientific notation, square that decimal number then square the power of 10. So 2.5 squared gives us 6.25. So now I need to square 10 to the power of 3, and here we're going to use our power of powers property, which says we can multiply those 2 exponents 3 times 2. So that gives a 6.25 times 10 to the 6th, and this is written in proper scientific notation.
So let's review multiplying and dividing in scientific notation. We talked about multiplication, division, and squaring. When you're multiplying, first you multiply the decimals then the powers of 10. In division, you divide the decimals then the powers of 10. And when you're squaring, you square the decimals then the powers of 10. There's a pattern there. And in each of these different operations, we used a different property of exponents. The product of powers, the quotient of powers, and the power of powers.
And remember in scientific notation, there can only be a single non-zero digit to the left of the decimal. So if you move your decimal to the left, increase your power of 10. If you move your decimal to the right, decrease your power of 10. Well, thanks for watching this video on multiplying and dividing in scientific notation. Hope to see you next time.