Exponentiation warmup

Exponentiation warmup with tree based model

Hi Sal.

>> Hey Brit.

>> How are you?

>> Good, it looks like you have a game
going on here.

>> Not a game, yeah, kind of a challenge
question for you.

What I did is I put one grain of rice in
the first square.

>> That's right.

>> There's 64 squares on the board.

>> Yep.

>> And in each consecutive square I
doubled the amount of rice.

>> Mm-hm.

>> How much rice do you think would be, on
this square?

>> On that square.

So let me, let me think about it a little
bit.

Actually I'm going to take some [SOUND].

So here you have 1.

And we multiply that times 2.

So this is going to be 2 times.

2 no, no 2 times 1, what am I doing?

Now this is 2 times that 1, so this is 2
times 2.

Now this 2 times that so this is okay,

we're starting to get a lot of 2s here,
multiplying them together so this is 2

times 2, I'm trying to write sideways
times 2.

So this one is going to be five 2s
multiplied together.

This is going to be six 2s multiplied
together.

This is gonna be seven twos multiplied
together.

>> Mm-hm.

>> Eight 2s multiplied together.

So nine 2s, 10, 11, 12, 13.

So 2, all, all of this stuff multiplied
together 8,192 grains of rice.

Is what we should see right over here.

>> You know I had fun last night, I was up
late, but there you go.

>> Did you really count out 8192 grains of
rice.

>> More or less.

>> Okay let's just say you did.

>> What if we just went four steps ahead,
how much rice would be here?

>> Four steps ahead, so we're gonna
multiply by 2, multiply by 2 again.

Then multiply by 2 again, then multiply by
2 again.

So this number times, let's see 2 times
two is 4, times 2 is 8 times 2 is 16.

So it's gonna get us like a 120, like
120,000 or around there.

>> 131,672.

You had a lot of time last night.

>> We're not even halfway across the board
yet.

>> We're not.

>> We, I mean this is, this is a lot of,

that's a lot of rice there, that, you
could throw a party.

>> What about the last square, this is 63
steps.

>> We're gonna take, we're gonna take 2
times 2, and we're gonna do 63 of those.

So this is going to be a, a huge number
and

actually, it would be neat if there was a
notation for that.

>> I, I didn't, I didn't count this one
out, but

it is the size of Mount Everest, the pile
of rice.

>> Hm.

>> And it would feed 485 trillion people.

>> But I have got a question.

I mean [INAUDIBLE] this was a little bit
of a,

of a pain, for me to write all of these
twos.

>> So is this.

>> If I were the mathematical community.

>> Mm-hm.

>> I would want some type of notation.

>> You kind of got on it here.

I, I like this dot, dot, dot and the 63.

This, you know, I understand this.

>> Yeah, you could understand this.

But, this is still a little bit.

Little bit too much, what if instead you
just wrote.

>> Mathematicians love being efficient
right they want, they're lazy.

They have things to do they have to go
home and count grains of rice.

>> Right.

>> [LAUGH] Yeah so that is 63 twos and
multiply them all together.

>> This is the first square on our board,
we have one grain of rice.

And when we double it, we have two grains
of rice.

>> Yup.

>> And we double it again, we have four,
and I'm thinking,

this, this is similar to what we were
doing, it's just represented differently.

>> Yeah, well I mean this one, the one you
were making, right, every time you,

you're kind of adding these more, these
Popsicle

sticks you're kind of branching out, you
know, one.

Popsicle stick now becomes two popsicle
sticks, and then you keep doing that.

One Popsicle stick becomes 2, but now you
have 2 of em.

So here you have 1, now you have 1 times
2.

Now each of these two branch into 2, so
now

you have 2 times 2 or you have four
Popsicle sticks.

Every stage, every branch you're
multiplying by 2 again.

>> I, I just basically continue splitting
just like the tree does?

>> Yup.

Now, I can really see what 2 to the power
of 3 looks like.

>> And, that's what we have here.

1 times 2 times 2 times 2, which is 8.

This is 2 to the third power.

>> And, when I see 2 to the power of
something, let's just say n.

>> Mm-hm.

>> N could also be number of steps up this
tree.

I could think about it that way.

>> Yeah, you could do, I guess one way to
think about it is

how many times you branched, but that one,
that tree there's actually even more.

>> I don't think this counts because,
again,

this branch is four times of each branch.

>> Well, I guess, why not.

Well.

I mean, its It's different, it's not gonna
be 2 anymore.

So this, the first one we haven't branched
yet.

This is gonna be 4 to the 0 power, you've
had no branches yet.

This you branched once, so this is 4 to
the 1st power, you have 4 branches now.

>> Oh, I like this.

>> And now each of those,.

So now you've branched twice, so this is 4
to the second power.

So yeah, the base, or what is called the
base, when

you're dealing with an exponent is four
right over here, this

is how many times, how many new branches
each of the

new branches turn in to at each of these
new junctions.

>> Let's call them junctions.

>> Junctions.

You haven't branched yet, here you branch
once and here you branch.

>> This is interesting.

This is also why when I look at a tree,

you know, there's thousands of leaves, but
just one trunk.

And when you actually go up, and you look
inside

the tree, it only branches you know, three
or four times.

>> And that shows the power of exponential
growth.

>> Yes.

[LAUGH]