WEBVTT

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Now that we've seen
that as we take i

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to higher and higher powers,
it cycles between 1, i,

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negative 1, negative i, then
back to 1, i, negative 1,

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and negative i.

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I want to see if we can
tackle some, I guess

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you could call them,
trickier problems.

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And you might see these surface.

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And they're also
kind of fun to do

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to realize that you can use
the fact that the powers of i

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cycle through these values.

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You can use this to really,
on a back of an envelope,

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take arbitrarily
high powers of i.

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So let's try, just
for fun, let's

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see what i to the
100th power is.

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And the realization here is
that 100 is a multiple of 4.

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So you could say that this
is the same thing as i

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to the 4 times 25th power.

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And this is the same thing, just
from our exponent properties,

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as i to the fourth power
raised to the 25th power.

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If you have something
raised to an exponent,

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and then that is
raised to an exponent,

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that's the same thing as
multiplying the two exponents.

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And we know that
i to the fourth,

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that's pretty straightforward.

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i to the fourth is just 1.

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i to the fourth is
1, so this is 1.

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So this is equal to
1 to the 25th power,

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which is just equal to 1.

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So once again, we use this
kind of cycling ability of i

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when you take its
powers to figure out

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a very high exponent of i.

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Now let's say we try something
a little bit stranger.

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Let's try i to the 501st power.

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Now in this situation, 501,
it's not a multiple of 4.

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So you can't just
do that that simply.

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But what you could do,
is you could write this

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as a product of two
numbers, one that

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is i to a multiple
of fourth power.

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And then one that isn't.

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And so you could rewrite this.

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500 is a multiple of 4.

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So you could write this as
i to the 500th power times i

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to the first power.

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Right?

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You have the same base.

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When you multiply,
you can add exponents.

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So this would be i
to the 501st power.

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And we know that this
is the same thing

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as-- i to the 500th power
is the same thing as i

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to the fourth power.

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4 times what?

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4 times 125 is 500.

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So that's this part right
over here. i to the 500th

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is the same thing as i to the
fourth to the 125th power.

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And then that times
i to the first power.

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Well, i to the fourth is 1.

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1 to the 125th power
is just going to be 1.

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This whole thing is 1.

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And so we are just left
with i to the first.

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So this is going
to be equal to i.

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So it seems like a really
daunting problem, something

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that you would have
to sit and do all day,

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but you can use this cycling
to realize look, i to the 500th

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is just going to be 1.

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And so i to the 501th is just
going to be i times that.

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So i to any multiple of 4--
let me write this generally.

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So if you have i to any multiple
of 4, so this right over here

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is-- well, we'll just restrict k
to be non-negative right now. k

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is greater than or equal to 0.

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So if we have i to any
multiple of 4, right over here,

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we are going to get 1, because
this is the same thing as i

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to the fourth power
to the k-th power.

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And that is the same thing
as 1 to the k-th power,

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which is clearly equal to 1.

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And if we have
anything else-- if we

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have i to the 4k plus 1 power,
i to the 4k plus 2 power,

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we can then just do this
technique right over here.

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So let's try that with a
few more problems, just

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to make it clear that
you can do really,

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really arbitrarily crazy things.

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So let's take i to
the 7,321st power.

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Now, we just have
to figure out this

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is going to be some multiple
of 4 plus something else.

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So to do that, well, you could
just look at it by sight,

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that 7,320 is divisible by 4.

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You can verify that by hand.

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And then you have
that 1 left over.

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And so this is going to
be i to the 7,320 times

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i to the first power.

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This is a multiple of 4-- this
right here is a multiple of 4--

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and I know that because
any 1,000 is multiple of 4,

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any 100 is a multiple of 4,
and then 20 is a multiple of 4.

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And so this right over
here will simplify to 1.

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Sorry, that's not i
to the i-th power.

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This is i to the first power.

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7,321 is 7,320 plus 1.

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And so this part right over
here is going to simplify to 1,

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and we're just going
to be left with i

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to the first power, or just i.

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Let's do another one.

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i to the 90-- let me try
something interesting.

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i to the 99th power.

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So once again, what's
the highest multiple

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of 4 that is less than 99?

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It is 96.

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So this is the same thing
as i to the 96th power times

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i to the third power, right?

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If you multiply these, same
base, add the exponent,

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you would get i
to the 99th power.

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i to the 96th power, since
this is a multiple of 4,

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this is i to the fourth, and
then that to the 16th power.

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So that's just 1 to the
16th, so this is just 1.

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And then you're just left
with i to the third power.

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And you could either remember
that i to the third power

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is equal to-- you
can just remember

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that it's equal to negative i.

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Or if you forget that,
you could just say, look,

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this is the same thing
as i squared times i.

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This is equal to
i squared times i.

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i squared, by definition,
is equal to negative 1.

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So you have negative 1 times
i is equal to negative i.

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Let me do one more
just for the fun of it.

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Let's take i to the 38th power.

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Well, once again,
this is equal to i

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to the 36th times i squared.

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I'm doing i to the 36th
power, since that's

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the largest multiple
of 4 that goes into 38.

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What's left over is this 2.

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This simplifies
to 1, and I'm just

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left with i squared, which
is equal to negative 1.