Okay. We've covered a lot of mass so far, we've done fractions, we've worked with multiplying, dividing, adding, and subtracting them. Let's try something new. Have you ever wondered why your skin cells get replaced? The skin cells on the top of your hand end up getting replaced, eventually they die off and they just fly away. You don't ever see them disappear, but they get replaced by newer cells underneath the top layer of your skin. So, here's what it would look like for one cell to replicate. So, let's start with one cell and see what happens. One cell is going to split into 2 cells, it's going to double, have 1 cell and then another cell. Then, each of these cells are also going to double, this one will make 2 cells and this one will make 2 cells. And then these four cells will also split. Each one will make two new cells. So, here's my question. How many cells will there be after seven rounds of division? By the end of the segment, you'll be able to answer this question. We're going to figure this out using exponents. Okay. This is just one cell. This is before any rounds of division. So, I'm going to mark that off with a 0. This is after the first round of division, I have 2 cells. And then, after the second round of division, I have 4 cells. And then, after the third round of division I have 8 cells. I know my cells are doubled every time, so I'm going to use a base of 2. Okay. I know I started with 1 cell and that cell doubled, so I have 1 times 2. Then, each of these cells also doubled, so I have 2 cells doubling or 2 times 2. Then I have 4 cells. Each of these 4 cells also doubled, so I have 2 times 2, those are my 4 cells. And then, they're going to double. There's got to be an easier way to write this. We're going to actually do this using exponents. Notice that all of my multiplication has a first doubling, I have 2^1 or 2 cells. After the second doubling, I have 2^2 or 4 cells. And after three rounds of doubling, I have 2^3 or 8 cells. And after three rounds of doubling, I have 2^3 or 8 cells. Repeated multiplication is represented by an exponent. It's very similar to repeated addition like 3+3+3+3. I know I have four threes or 43. We use multiplication to represent repeated addition, we're going to use exponents to represent repeated multiplication. So, whenever I have a repeated multiplication, I can write the base raised to an exponent. So whenever I have repeated multiplication, I can have the base raised to an exponent.
Okay. Let's try your understanding of this with a quiz. You can put your basis here and you can put your exponents here. Remember, exponents are written as superscripts. They're written just above the base.
For the first one, you should have a base of 4, since you have a repeated multiplication of 4, and you have it repeated one, two, three times. So we have Here I have a repeated multiplication of one, two, three, four, five times. Here I just have one 7, so I'm going to put 7 as my base. I don't have any repeated multiplication. I just have the 7 once, so my exponent is a 1. Here I have the 3 as my base. And I have the 3 repeated 7 times, so 7 is my exponent. Nice work on that quiz.
Okay, lets look more closely at the meaning of an exponent. Its easy to think of 2^3 as 6, but we really know this isn't true. I know exponents represent repeated multiplication, so I should have. 2 2 that students make throughout algebra. You have to train your eye and train your mind to think about what this really means. Go slow and be careful. This is how you can get better at math, taking your time and asking yourself, what does this mean?
Okay, let's try this out with a quiz. Which of these answers has the correct base and exponent for 18? Is it 2^4, 3^6, Choose the best answer.
Okay, the real answer is none of these. For 2 ^ 4, I know that's 2 2 2 2. I've repeated mulitplication 4 times, so I get 16. That's definitely not 18. For 3 ^ really 9 9 9 or 729, that's definitely not 18. 6 ^ 3 is 6 6 6, so I get 216, here I just have 9 ^ 2, so 9 9, I have the repeated multiplication of 9, well that makes 81, 81 isn't 18 even though they both have digits of 8 and 1. For this last one, I just have 1, 18 times. That's a lot of 1's. But I know 1 1 1 1 is always going to be 1. So this answer is going to equal 1. So that's not it either. This is why it's just important to think about what an exponent means. Most of the math you do on a daily basis involves multiplication. That's why it's so easy to think of exponents as multiplication. We have to slow down and remind ourselves that 2 ^ 4 isn't really 8. It's really 16. Just like 9 ^ 2 is really 81.
Alright. Let's look at some exponent patterns now to figure out what goes on with other types of exponents. We've seen them increase. Well, what happens when they decrease? Okay, here I have 3^3 which is really just 333, or 27. Here, I have just have 3. Based on this pattern, what do you think 3^0 would equal? Think about what's happening to our answer as we go down. You can put your answer in this box here.
Okay. The answer should be 1. Here's how we figure that out. I know as I go from 27 to 9, I divide by 3. And then to get to 9 to 3, I divide by another 3. So to find 3 to the 0, I need to take my number, and divide it again by 3. 3 / 3 makes 1. That's how I know 3^0=1. It turns out this is true for any power of I know as the exponent decreases, this number also decreases. To get from 8 to 4, I divide it by 2. This should make sense because I know exponents represent repeated Multiplication. So to drop one exponent, I need to undo multiplication. I need to do Division. And then to get from from 2 to 2 to the 0, I have to divide by exponant 1, I have to undo Multiplaction, so I do Division, and I get 1.
Okay. But let's not just stop at zero, let's keep going. What happens if we have negative exponents? I would have 3^-1, so I know I have to take one and divide it by three. I remember dividing by a number is the same thing as multiply by the reciprocal. So, 1 divided by 3 is the something as 11/3. If I had 3^-2, I'd need to divide this again by 3. I know dividing by 3 is the same thing as multiplying by a reciprocal. So, I have 1/3 and then, times 1/3 and I get 1/9. The same is true for 3^-3. I would take 1/9 and divide it by 3. When I divide 1/9 by 3, it's the same thing as multiplying by the reciprocal. So, I have
Now that you've seen negative exponents, let's try some of your own. What would be the answer to these negative exponents? You could put your answer in these boxes. Remember to put the number, the fraction bar, and then a number.
Alright. So, for the first one, 4^-1, I know right above it, I have 4^0 or I have to divide by 4. 1 divided by 4 is 1/4. For the second one, 4^-2, I just take 1/4 and divide it by 4. Remember, dividing by a number is the same thing as multiplying by the reciprocal. 1/4 times 1/4 or 1/16. And finally, 4^-3, so I need to take 1/16 and divide it by 4. And remember, dividing by a number is the same as multiplying the reciprocal. So, I have 1/161/4 or 1/64. Nice work on that quiz.
With negative signs, reciprocals, and exponents, you want to be very careful. Math requires a keen attention to detail. Let's look at a couple scenarios when we'll be working with negatives so that way you can see what I mean. We have to be careful. Here I know it's just 5^2 or 5 times 5, so I get 25. For -5^2, I have the whole quanitity of -5^2, so I write the whole thing in parentheses twice. So I have -5 times -5. I know a negative times a negative is a positive, so I get 25. Here, I have to remember back to my order of operations. Do exponents, or does subtraction, take precedence? I know exponents come first. Here, I have a choice. I could have the 5 be negative. And the I could square it or I could square the 5 and then take the negative. Remember back from the order of operations. We know exponents come first. So I have 5 ^ 2 and then I'll have it multiplied by -1. Oftentimes we don't show the multiplication here but that's really what we mean. It's -1 times 5. This is why we have to be really careful about what we mean in Math. That's why we, a lot times, we use parenthesis. We want to know if we're squaring 5, negative 5, or, if we're squaring a number, and then multiplying it by negative 1, they're all different. So for my answer, I have negative 1 times 25, or negative 25. And finally, 5 to the negative 2. Remember, with negative exponents, we really do the reciporical. So I'm going to have 1 over 5, and that's going to be squared. So I have 1/5 times
Okay. Let's try a quiz on this information. Okay, which of the following are equal to -9? Is it ^-2, -3^2, the quantity -3^2, 1/3^2, -1/3^-2, or -1/3^2. Okay, I know this looks like a lot of negatives, but if you work through it slowly, you can probably figure out what everything means, and get to the right answer. This is a tough quiz, so take your time and work through it slowly.
Okay. We're going to cover one last thing with exponents, and it has to do with squares. In particular, I mean squaring numbers, like 4^2 and 5^2. Specifically, I want to know if 4^2+5^2 is the same thing as the quantity of 4+5^2. I'm going to draw some squares to help us out. This square's going to have sides of of 5. This square's going to have sides of I can see the side with 4 and than 5. It will look like this, this side is going to be 4 and this side is going to be 5. I've kind of exaggerated it but its okay. This side is also going to be 4 and this side is going to be 5.
Okay. What I want you to do is come up with the area of each of these squares. So, find the area of this square, this square, and then this square has four different regions. I want you to find the area of each one. So, for example, to find the area of this part, I know that this side is 5, and I know that this side is the same as this side, which is same as this side. So. 5 x 5, or 25. You multiply the sides of a rectangle or a square to get the area. Go ahead, fill in these areas.
I knew this was 16 because 44 is 16, 55 is 25. This one had an area of 20, 54 is top side, so I should have 44 or 16. And then, for over here, I have 4 for this piece, and 4 for this piece, 5 for this piece, and 5 for this piece. 45 is 20.
I know 4^2 is really 16 and 5^2 is really total area of 41. For the blue side, I have 20+16+20+25. When I add all these areas together, I get an area of 81. This makes sense because I know if I would have just combined the sides, I would have had 9. And then I would have squared it. 9^2 is 81. I know these two numbers are not equal. We want to be really careful. Exponents cannot distribute over sums or differences. We can't just say 4+5^2 is the same thing as This is a key concept that we will come back to later. I just want to preview it for now and give you an idea of area models. This is going to help us problem solve. Now, let's have you practice with some exponents.
Now that we know a little bit more about a square or a 2 exponent, let's try solving some problems. Try solving these problems and put your answer in each of these boxes. Good luck.
For the first problem, we have the quantity, -5^2. We have -5 twice and -5 times -5 is +25. For the second problem, we have the negative of 8^0. Remember, when I have a negative sign out in front, it's a -1. So -1 times 8^0. Any number raised to the zero power is just one. So, we have -1 times 1, -1 is our answer. Number 3 is pretty tricky. We have 1^100. It might be tempting to write 100 as the answer but we really know that this is 1 times 1 times 1, a 100 times. This is a lot of ones. But we know when we multiply all these ones together, we'll just wind up with 1. For number 4, we have the negative of 5^2-2^2. This negative sign out in front is really -15^2, so I'm only needing to square, the difference between #1 and #4. In #1, we had the quantity negative 5^2. Here, we only have the negative of 5^2. So, we just square the 5. 5^2 is 25 times -1 is -25. Then, I have a -4 on the end. -25-4, well, I have -25 and -4. So, that's more negatives. I have -29. And finally, for number 5, we have the quantity, -4+7^2. Remember, we can't distribute the square over the sum. We need to add these numbers together first, then do the square. -4 and got a least half of these right, excellent work. Some of these were a little bit tricky, so it's okay if you didn't get all of them right, but I'm sure you will next time.
Okay, let's get back to our original problem on cellular division. I wanted to know how many cells would there be after 7 rounds of division. Remember I started with one cell and that split into 2 cells. You can draw a diagram to model this out, or you can just use an exponent. So, if I started with one cell and had 7 rounds of division, each cell doubled, how many cells would there be? You can put your answer in this box.
Okay, I know my cell is doubling everytime so I have 2. And then there's 7 rounds of doubling, so I have 2^7. Here are my seven 2's I'm going to group them and then I'm going to group these. So I have 16 8. I know 8, 8 6 is 48, so I put an 8 and carry the 4. 8 1 is 8 + 4 is 12 so 128. So if I start from 1 cell. After 7 rounds of division, there would be
Okay. Let's put everything you know about exponents together and answer this question. A cell divides into 4 cells after 2 hours. If each new cell also divides into 4 cells after 2 hours, how many cells are there after one day? You're going to need to think about time and how many rounds of division you'll have. Remember, you're going to start with one cell and then it's going to split into four cells. So, after one round you'll have four cells. Then each of these cells is going to split into four cells. You can put you answer in this box here.
Alright. I need to think about how many rounds of divisions I'll have. I know my cells are going to replicate over one day. One day is 24 hours. I know my cell divides every two hours so I can take 24 hours and divide it by 2 hours, that would give me 12. This is the number of rounds of division I'll have. My cells are quadrupling each time after a round of division, I went from 1 cell to 4 cells. So remember, this cell will split into 4 more cells, this one would, too, and this one would, too, and so would this last one. I went from 1 to 4 to 16. I'm quadrupling the number of cells I have. So, my base should be 4 and I have 12 rounds of division, 4^12. And I get a answer of 16,770,216. That's a ton of cells. We're going to figure out an easier way to write this number in the next section. Great job, for now, if you got that right.
Now it's time to practice what you learned and try evaluating some exponents. You can put your answers in these boxes. Good luck.