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Multiplication and Division in Scientific Notation

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Today we're going to talk about multiplying and dividing numbers in scientific notation. Remember, scientific notation is just a shorter way of writing a number with a lot of digits. So numbers that are really big or numbers that are really small. So we'll start by reviewing the rules for writing a number in scientific notation and then we'll do some examples.

So a number written in scientific notation follows this general form. I've got some number, a, that's a decimal, times a power of 10, so 10 to some exponent. So for example, I could have 4.362 times 10 to the 12th. And there's two rules that you have to know for writing a number in scientific notation. The first one is you can only have one non-zero digit in front of your decimal, or to the left of the decimal. But you could have any number of digits after your decimal, or to the right of the decimal.

So for example, if I had the number 506.32 times 10 to the seventh, even though it looks like the general form, it's not in proper scientific notation, because I need only one number to the left of the decimal. So I need to move my decimal over to right after the 5. So if I move it over to after the 5, I'm moving it over twice. And we remember that moving your decimal to the left is going to increase our exponent. So my exponent is going to increase by two. So this is going to become 5.0632 times 10 to the ninth.

Another example would be a 0.049 times 10 to the negative 6. Again, this is not in proper scientific notation form because I don't have a non-zero digit to the left of my decimal. So here I'm going to have to move my decimal over to the right so that it comes right after the 4. So this will be 4.9. And since I moved my decimal to the right, I'm going to be decreasing my exponent. So instead of negative 6, this will become 10 to the negative 8. All right, so now that we reviewed the rules for writing a number in scientific notation, let's do some examples, multiplying, dividing, and squaring these kinds of numbers.

So for our first example, we're going to multiply 1.04 times 10 to the eighth times 2.6 times 10 to the negative fifth. When we're multiplying numbers in scientific notation, we can multiply them in two different parts. And that's because multiplication is commutative. So I can start by multiplying my decimal numbers together. And then I can multiply my powers of 10 together. So 1.04 times 2.6 is going to give me 2.704. And then, when I multiply 10 to the eighth times 10 to the negative fifth, I'm going to use my product of powers property.

Remember, the product of powers property tells us that if the base is the same, then we can simply add our exponents and write it as one power of 10, or write it as one exponent. So our base is the same-- they're both a 10. And this is going to work always for scientific notation. We can always use the product of powers property, because our base is always going to be a 10 in scientific notation.

So I'm going to add my exponents, 8 plus negative 5, and that's going to give me 3. So this will be times 10 to the third. So this is my final answer, but I want to make sure that this is in proper scientific notation. And it is, because I have one number in front of my decimal and I can have any number of digits after, so this looks great.

Here's my second example. I've got 8.36 times 10 to the third over 3.2 times 10 to the fifth. This is a fraction, which means we're going to be dividing. So similar to our first example when we were multiplying, we're going to start by dividing our decimal numbers. And then we'll divide our powers of 10. So 8.36 divided by 3.2 is going to give me 2.6125. And now when I divide my powers of 10, I'm going to use my quotient of powers property. And that tells us that if our base is the same, then we can subtract our exponents and write it as one exponent.

So again, this is always going to work. This property will always work for scientific notation, because our base is always going to be a 10. It's always going to be the same. So 3 minus 5 is going to give me a negative 2. So this is going to be times 10 to the negative 2 exponent. So again, I need to check and make sure that this is in proper scientific notation. And it is, we only have one number in front of the decimal. And we can have any number of digits after, so this is our final answer.

So for our last example, we're going to look at how to square a number written in scientific notation. So I've got 6.79 times 10 to the negative third squared. So we remember that squaring a number means that we're just multiplying it by itself twice. So one way to do this would be to multiply 6.79 times 10 to the negative third times 6.79 times 10 to the negative third. Just in the way that we did example one. But I'm going to show you how to do it by distributing your exponent.

So similarly to with multiplying and dividing, we're going to look at this in two pieces. First with the decimal, then with the power of 10. And we're going to distribute our exponent to both of these parts. So 6.79 squared is going to give me 46.1041. And then I'm going to distribute my 2 exponent to the 10 to the negative third. And to do this, we're going to use the powers of power property, which tells us that we can multiply our two exponents and create just a single power of 10. And this, again, always works for scientific notation because we're always using a base of 10.

So negative 3 times 2 is negative 6. So this will give me times 10 to the negative 6. All right, so here I'm running into a problem, because this is not exactly in scientific notation. I need to move my decimal so there's only one number in front of it. So I'm going to move my decimal over to the left. Moving your decimal to the left is going to increase your exponent. And since I moved it once, I'm going to increase it by one. So this is going to become 4.61041 times 10 to the negative 5. And this will be my final answer.

So here are the key points that we talked about today. Make sure that you get them in your notes if you don't have them already so you can refer to them later. So the first one is that when you're multiplying numbers that are written in scientific notation, you're first going to multiply your decimal numbers. And then using the product of powers property, you're going to add your exponents. The second key point was that when you're dividing numbers in scientific notation, you're again first going to divide the decimal numbers then use the quotient of powers property to subtract your exponents. And the last key point is that when you're squaring a number in scientific notation, you're first going to square the decimal number then use the power of powers property to multiply your exponents.

So I hope that these key points and the examples that we did helped you understand a little bit more about multiplying, dividing, and squaring numbers written in scientific notation. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.