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Addition and Subtraction in Scientific Notation

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Today we're going to talk about adding and subtracting numbers that I've written in scientific notation. Remember, scientific notation is just a shorter way of writing a number that has a lot of digits, so numbers that are really big or 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 let's review how to write numbers in scientific notation. In scientific notation, we have some decimal number a multiplied by some power of 10 with any positive or negative exponent n. So for example, 3.54 times 10 to the 12th is written in scientific notation.

Now, there's a couple of rules when you're writing in scientific notation. The first is that you can only have one non-zero digit to the left of your decimal and you could have any number of digits to the right of your decimal. So for example, 346.2 times 10 to the 4th is not in proper scientific notation, because we, again, have more than one number to the left of our decimal.

So to change it to proper scientific notation, I can move my decimal over two times so it's right behind the 3. And I know that moving my decimal to the left is going to increase my exponent. So by moving my decimal over to the left two times, this will become 3.462. And my exponent is going to increase by 2-- so times 10 to the 6th.

For this example, 0.037 times 10 to the 8th is not in scientific notation, because we have a 0 in front of our decimal. We need a non-zero digit, one non-zero digit. So now I want to move my decimal over two to the right so that is right behind the 3.

And I know that moving my decimal to the right is going to decrease my exponent. So when I move my decimal over, this is going to become 3.7. And my exponent again is going to decrease by two-- so times 10 to the 6th.

So here's my first example. I've got two numbers written in scientific notation that I want to add. And I notice that they both have the same power of 10. So when you have two numbers in scientific notation that you want to combine and they have the same power of 10, you can simply add or subtract your decimal numbers and then keep your power of 10.

Another way to think about this is we can combine our decimal numbers. Here we're going to add. And we're going to factor out our similar or common power of 10. So before we solve it this way, it let me show you how that works.

So I'm going to first convert this into standard form. 6.32 times 10 to the 5th is going to give me 632,000. 1.809 times 10 to the 5th is going to give me 180,900. Adding those together, I'm going to get 812,900.

Now, if I were to convert that into scientific notation, I'd need to move my decimal over five times to the left. So that's going to give me 8.129 times 10 to the 5th. All right, so if we go back to this method of simply adding our decimals and factoring out our power of 10, I know that I should get the same answer.

So adding my decimal numbers together, 6.32 plus 1.809 gives me 8.129. And I have my same power of 10, which I factored out-- so 8.129 times 10 to the 5th. So I see that by keeping my power of 10 or factoring it out and adding my decimals, I arrived at the correct answer.

So here's my second example. I've got two numbers written in scientific notation that I want to add. However, I notice that they don't have the same power of 10, which means I can't use my previous method yet of just factoring out the power of 10. I need to make sure that they have the same power of 10 first.

So I'm going to go ahead and change this number to have the power of 10 of 10 to the 14th so that they have the same power of 10. I'm going to do that by moving my decimal point to the left so that it increases from 10 to the 12th to 10 to the 14. So moving my decimal point over to the left two times, I see that I'm going to need to add a placeholder of 0 here. So this is going to become-- this number in scientific notation will become 0.05362 times 10 to the 14. And here, this number does not need to change, because I now have a common power of 10.

So now I can factor out my power of 10. And so that will look like 0.05362 plus my decimal number for my second number, 4.63. And now since my power of 10 is the same here and here, I can write it as times 10 to the 14. Now all I need to do is add my two decimal numbers. So that's going to give me 4.68362-- and I've got my common power of 10-- times 10 to the 14th.

So here's my third example. I've got 6.9883 times 10 to the negative 2nd minus 3.4 times 10 to the negative 4. So I see that my powers of 10 are not the same, which means that I cannot simply factor out a power of 10 and subtract my decimals.

So I'm going to change one of these numbers so that it has the same power of 10. And then I can go ahead and just factor it out and subtract my decimals. So again, it doesn't matter which one you choose. I'm going to make this number to have a power of 10 of negative 2. I'm going to do that by moving my decimal point over to the left two times.

I'm moving it over to the left two times because I know that moving a decimal to the left is going to increase it. And I need to increase it twice to go from negative 4 to negative 2. So when I'm moving my decimal point over twice, this number becomes 0.034 times 10 to the negative 2.

So now that my powers of 10 are the same, I can go ahead and just factor out the power of 10 and subtract my decimals. So this will look like 6.9883 minus 0.034-- factor out my power of 10-- 10 to the negative 2. Subtracting my decimal, 6.9883 minus 0.034 will give me 6.9543 times 10 to the negative 2ne.

So let's review our key points from today. Make sure that you get these in your notes if you don't have them already so you can refer to them later. So the first is that when you're adding and subtracting numbers that are written in scientific notation, if the powers of 10 are the same, you can just add or subtract your decimal numbers. And then you're just going to keep the same power of 10.

And the second key point is that if the powers of 10 are not the same, then you can change the exponent for one of the numbers that's written in scientific notation by moving the decimal either to the left if you want to increase the exponent or by moving it to the right if you want to decrease the exponent. So I hope that these notes and these examples helped you understand a little bit more about adding and subtracting numbers in scientific notation. Keep using your notes and keep on practicing, and soon you'll be a pro Thanks for watching.