 So we're going to talk about scientific notation. First thing we want to talk about is just a definition for scientific notation. And anything that's written in scientific notation will be written in the form of a number a times 10 to the n power. Some things that you need to know about the a and the n. First off, a must be a number between 1 and 10. Very important. The second thing, the n. So the n is an integer. And that integer can be either positive or negative. If n is positive, that means that the value here in front, the a value, is going to get larger. If the n value is negative, then the a gets smaller. So very important to remember that. So scientific notation is really used when you're working with very large and very small numbers. For example, if you have 5 times 10 to the fourth, first off, that is truly in scientific notation. 5 is the number between 1 and 10. And 10 to the fourth, the fourth is an integer. So in order to change this into what we call standard notation, what you're going to do first is always take the a value and bring it with you. So a is 5. 5 is simply going to stay with you. And then we're going to go ahead and put the decimal in place. There's not a decimal currently, so I know that it's going to go after the 5. And then we're going to use the integer, the power on 10, which is 4, to tell us what to do. Just remember that if it's positive, then this a value that I'm starting with should get larger. Well, in order for it to get larger, obviously the decimal has to move to the right. The number, the integer here, the power tells you how many places to move the decimal. So I know it's getting larger. I know I have to move four places, so that means I have to move to the right four places. So 1, 2, 3, 4. All of the places that are empty will be filled with z-rays. So this becomes 50,000. And that's in what's called standard form. Now let's try another one. What if you have 5 times 10 to the negative fourth? Again, it's written in scientific notation. The first thing that you want to do is to keep your a value, which is 5. Bring that with you. Go ahead and put the decimal in place, which would be currently after the 5. And then look at your power. The power is negative 4, which is telling you that the number should be getting smaller. In order for the a to get smaller, the decimal is obviously going to have to move to the left. And so the 4 tells me to go four places. So I'm simply going to take and move the decimal four places to the left. And again, all of the holes are filled in with z-rays. So the new number becomes 0.0005. So we've just practiced going from scientific notation to standard notation. So now we're going to try to go backwards. And I don't want to teach you any new rules. I want to use what we already have learned to be able to do this. So now we're going to try to go from standard to scientific. So we start with 4,563. The first thing that we need to do is to make that become a number between 1 and 10. Remember, the a value has to be between 1 and 10. So currently, the decimal is located after the 3. So right there. So I'm going to continue to move the decimal place until I get a number that's between 1 and 10. If I move it one time, that would now read 456.3, which is not between 1 and 10. If I move it a second time, that would become 45.63, which is still not between 1 and 10. If I move it one more time, I get 4.563, which is between 1 and 10, which is what I'm looking for. So I know that my a value is going to be 4.563. And I know in scientific notation, I'm always going to have times 10 to a power. All I need to do is figure out what that power is. And it's very simple. Always, when you're working these problems, think about going from scientific to standard, and you'll never have to remember a bunch of rules. So in scientific notation, this is my value. In standard, it was 4,563. What happened to that a? Did a get larger? Did a get smaller? Going from scientific to standard, it got larger, meaning that the power is going to have to be a positive power. So all I need to know is how many times I moved, which was three times. So it's going to be 10 to the third power. So again, always consider going from scientific to standard, and that'll help you say you don't have to have two rules. For the second one, same idea. We have 10,000. The decimal is currently located after the last zero. And we know we have to move the decimal until we have a number between 1 and 10. So I'm going to keep moving it until I have it. Here, I'm at 10. I need to go one more. So I end up with 1.000. Well, all of these zeros after the decimal place are not important. I can drop them because there's no other number over here, so I'm just going to call it 1. So I have a number between 1 and 10. And then, I know with scientific notation, it's going to be times 10 to some power. All I have to do is figure out what that power is. So first off, how many places did we move the decimal? So 1, 2, 3, 4 places. So I know it's going to be a 4. All I have to do is figure out if it's going to be a positive 4 or a negative 4. Well again, easy way to do this. Always look at the scientific notation and then go to standard. In scientific notation, it's a 1 in front. In standard, it's 10,000. It got larger, so since it got larger, I know that it has to be a positive number. Still the same idea, going from standard to scientific. This time, the numbers are a little different. So in number 3, we have 0.00391. And again, the first thing that we want to do is to get that number between 1 and 10. So I'm going to move the decimal place. 1, 2, 3 times because that would turn into 3.91, which is definitely a number between 1 and 10. And then I know in scientific notation, it's going to be times 10 to a power. Well, let's start with this, starting with how many times I moved the decimal. So I moved it 1, 2, 3 times, so I know it's going to be a 3. And then all I have to decide is, is that 3 positive or is it negative? Again, very simple. Look at the scientific notation. Always look at scientific notation. And then go to standard. Did the number get larger or smaller? It got smaller. So in order for it to get smaller, it would have to be a negative power. So 3.91 times 10 to the negative 3. And our last example going from standard to scientific notation is 0.57. So how many times do I have to move the decimal to get it to be a number between 1 and 10? Only once. So that would become 5.7. And again, in scientific notation, we always have times 10 to a power. We just have to figure out that power. Well, first off, we moved it once, so I know that that power is going to have a 1. I just need to figure out if it's positive or negative. So again, looking at the scientific notation, the a value in scientific notation, going back to the original number, did it get larger or smaller? The number got smaller, so I know that it's going to be a negative.