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Contents

- 1 cellular division
- 2 writing exponents
- 3 writing exponents
- 4 meaning of an exponent
- 5 exponents
- 6 exponents
- 7 exponent patterns
- 8 exponent patterns
- 9 the negative pattern
- 10 negative exponents
- 11 negative exponents
- 12 the meaning of a negative power
- 13 working with negatives
- 14 squares
- 15 area of squares
- 16 area of squares
- 17 Unequal Squares
- 18 Working with squares and powers
- 19 Working with squares and powers
- 20 cellular division solved
- 21 cellular division solved
- 22 quadrupling cell
- 23 quadrupling cell
- 24 Exponent Practice

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 4*3.
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 3*3 3, 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 1*1/3.*
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 1/16*1/4 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 4*4 is 16, 5 5
is 25. This one had an area of 20, 54 is*
top side, so I should have 4

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
-1*5^2, 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.