Hi, and welcome. My name is Anthony Varela. And today, I'd like to talk about addition and subtraction with scientific notation. So first, we're going to review scientific notation on how to write numbers in proper scientific notation. And that involves moving decimals and adjusting exponents. So we're going to practice doing that, because that's going to be really important when it comes time to adding and subtracting numbers in scientific notation.
So let's review scientific notation. Scientific notation puts numbers in this form-- a times 10 to the power of b. So it's a decimal number times a power of 10. And we're dealing with positive powers of 10. We're talking about very, very large numbers. And when we're talking about negative powers of 10, that's very, very, very small numbers.
Now, there are some rules about what this number, a, can look like. a can only have a single digit to the left of the decimal. There can be as many digits as you want to the right of the decimal. That just means it's a more accurate number. But there can only be one digit to the left of that decimal, and that digit cannot be 0.
So let's take a look at this number here, 13.1 times 10 to the sixth. This is not written in proper scientific notation, because I see more than one digit to the left of the decimal. So what I have to do is I have to shift that decimal over. And I'm going to go to the left one place. So now I have 1.31. But now I have to adjust my power of 10 to keep this equal, and I'm changing my power of 10 from 6 to 7. And that's because whenever you move your decimal to the left, you increase your power of 10 in scientific notation.
Let's take a look at this number. Is this written in proper scientific notation? I see that there's only one digit to the left of the decimal, but it's 0, and that's the only number it can't be, right? So this is not written in proper scientific notation. I have to move my decimal, but I have to move it in the other direction. I have to move it to the right.
Now, if I move it to the right one place, I still have a 0 to the left of the decimal, so I have to actually move it over two places, so I have 5.2. But now I have to change my power of 5. And here, I am moving my decimal to the right, so I have to decrease my power of 10. But I moved it to the right twice, or two places, so I have to decrease my power of 10 by 2. So now I have 5.2 times 10 the third, and that's because when I moved my decimal to the right. I have to decrease my power of 10.
Well, now let's get into adding numbers written in scientific notation. So here, I have 5.0 times 10 to the second power. And I'm adding to that 1.7 times 10 the second power. And I'm going to go about this the long way, and then we're going to reveal a shortcut. So I'm going to expand my powers of 10. So I'm just writing this as 5.0 times 10 times 10 and 1.7 times 10 times 10.
Well, I know that 10 times 10 equals 100, so I'm going to write this as 500 plus 170. That's just taking 100 and multiplying it by 5 to get 500 and taking 100 and multiplying that by 1.7 to get 170. Well, now this is a pretty simple addition problem. I know how to add this. This equals 670.
Well, let's write 670 in scientific notation. This would be 6.7 times 10 to the second power. Now notice what I've really done here is I've factored out a common factor of 10 to the second power, which meant I can just add my two decimal numbers and multiply that by my power of 10.
So for common powers of 10, what we're going to do is add the decimals, 5.0 plus 1.7 that got us 6.7, and keep the power of 10-- 10 to the power of 2. And that looks like this, written very generally, where a and b are two decimal components up here, and then we're keeping our power of 10. Well, what happens, then, if our powers of 10 are not the same? So what are we going to do?
Well, taking a look at this problem, we have 6.2 times 10 to the second power and 2.8 times 10 to the third power. Notice that this does not fit our description for common powers of 10. We cannot simply add the decimals and keep the power of 10 or do that common factoring that we did before. But what we can do is we can rewrite one of these numbers to make our own common power of 10. And that would require moving our decimal and adjusting our exponent.
So it doesn't matter which number you choose to manipulate. You just need to pick one. And so what I'm going to do is I'm going to rewrite 6.2 times 10 to the second power. I'm going to rewrite that as 10 to the third power by moving my decimal.
So I had to move my decimal. Let's see. I wanted to increase my exponent, so that means I had to move my decimal place over to the left. So you've seen I've done that here. I have moved my decimal over one place to the left, so I have 0.62 times 10 to the third. I just followed that rule right here.
Well, now what do we notice? We have our common powers of 10. So we can actually follow this rule where we add our decimal components and keep our power of 10. So I need to add 0.62 to 2.8, and that gives me 3.42 and I'm keeping my power of 10. It's that simple. So when we have uncommon powers of 10, we need to choose one of those numbers in scientific notation, shift the decimal, and adjust the exponent to make a common power of 10. And then you can go back to following this rule.
Now, let's see how this works with subtracting two numbers in scientific notation. So here, we have two numbers in scientific notation we're going to subtract here, and we also have our uncommon powers of 10. So what does this mean? We have to choose one of the numbers, shift the decimal, and adjust the exponent to make our own common power of 10. And like I said before, it doesn't matter which number we choose. Just pick one.
So I'm going to rewrite 8.32 times 10 to the fifth. I'm going to rewrite that so that it's 10 to the power of 3. So I had to decrease my exponent, so that means I had to move my decimal to the right. And I decreased it by 2, so I have to move my decimal to the right two places. So that's how I got 832. So now I have common powers of 10. And here, I'm going to subtract my two decimals.
So this is the difference from addition. So really, what I'm going say here is for common powers of 10, we're going to combine the decimals. And that means if you're doing an addition problem, you're going to add the decimals. If you're doing subtraction, you're going to subtract the decimals.
So here, then, I'm just going to take 832 and subtract 7.1 and keep my 10 to the third-- my common power of 10. Well, 832 minus 7.1 is 824.9. Well, this is not written in proper scientific notation. I have more than one digit to the left of my decimal.
So once again, I have to shift my decimal and adjust my exponent. And I probably could have avoided this if I decided to manipulate this at the beginning. But like I said, it's not wrong to do it this way. So I need to shift my decimal-- let's see-- 1, 2 places to the left. So I'm going to increase my exponent by 2. So I have 8.249 times 10 to the fifth power.
So to review what we talked about today, adding and subtracting numbers in scientific notation-- if we have common powers of 10, we can just combine our decimal components and keep that power of 10. So this is how it's written when you're adding, and this is how it's written when you're subtracting. Here are our two decimal components, keeping that power of 10.
If we have uncommon powers of 10, we can create our own common powers. We just need to shift the decimal, adjust the exponent, and then we'll have our common powers of 10. And remember, when you're doing that, shifting the decimal to the left means increasing the power of 10, and shifting the decimal to the right is decreasing your power of 10. Well, thanks for watching this video on adding and subtracting numbers in scientific notation. Hope to catch you next time.
