0:00:00.520,0:00:02.810
Now that we've seen[br]that as we take i

0:00:02.810,0:00:06.740
to higher and higher powers,[br]it cycles between 1, i,

0:00:06.740,0:00:11.260
negative 1, negative i, then[br]back to 1, i, negative 1,

0:00:11.260,0:00:12.290
and negative i.

0:00:12.290,0:00:14.155
I want to see if we can[br]tackle some, I guess

0:00:14.155,0:00:15.780
you could call them,[br]trickier problems.

0:00:15.780,0:00:17.110
And you might see these surface.

0:00:17.110,0:00:18.526
And they're also[br]kind of fun to do

0:00:18.526,0:00:22.180
to realize that you can use[br]the fact that the powers of i

0:00:22.180,0:00:23.320
cycle through these values.

0:00:23.320,0:00:25.990
You can use this to really,[br]on a back of an envelope,

0:00:25.990,0:00:29.100
take arbitrarily[br]high powers of i.

0:00:29.100,0:00:31.650
So let's try, just[br]for fun, let's

0:00:31.650,0:00:35.310
see what i to the[br]100th power is.

0:00:35.310,0:00:39.280
And the realization here is[br]that 100 is a multiple of 4.

0:00:39.280,0:00:43.800
So you could say that this[br]is the same thing as i

0:00:43.800,0:00:47.467
to the 4 times 25th power.

0:00:47.467,0:00:50.050
And this is the same thing, just[br]from our exponent properties,

0:00:50.050,0:00:55.167
as i to the fourth power[br]raised to the 25th power.

0:00:55.167,0:00:57.000
If you have something[br]raised to an exponent,

0:00:57.000,0:00:59.090
and then that is[br]raised to an exponent,

0:00:59.090,0:01:02.300
that's the same thing as[br]multiplying the two exponents.

0:01:02.300,0:01:04.170
And we know that[br]i to the fourth,

0:01:04.170,0:01:05.420
that's pretty straightforward.

0:01:05.420,0:01:07.390
i to the fourth is just 1.

0:01:07.390,0:01:09.590
i to the fourth is[br]1, so this is 1.

0:01:09.590,0:01:12.300
So this is equal to[br]1 to the 25th power,

0:01:12.300,0:01:15.910
which is just equal to 1.

0:01:15.910,0:01:18.867
So once again, we use this[br]kind of cycling ability of i

0:01:18.867,0:01:20.450
when you take its[br]powers to figure out

0:01:20.450,0:01:22.672
a very high exponent of i.

0:01:22.672,0:01:24.880
Now let's say we try something[br]a little bit stranger.

0:01:27.730,0:01:31.200
Let's try i to the 501st power.

0:01:31.200,0:01:34.620
Now in this situation, 501,[br]it's not a multiple of 4.

0:01:34.620,0:01:36.310
So you can't just[br]do that that simply.

0:01:36.310,0:01:38.226
But what you could do,[br]is you could write this

0:01:38.226,0:01:41.500
as a product of two[br]numbers, one that

0:01:41.500,0:01:44.140
is i to a multiple[br]of fourth power.

0:01:44.140,0:01:45.580
And then one that isn't.

0:01:45.580,0:01:47.050
And so you could rewrite this.

0:01:47.050,0:01:50.390
500 is a multiple of 4.

0:01:50.390,0:01:56.000
So you could write this as[br]i to the 500th power times i

0:01:56.000,0:01:56.960
to the first power.

0:01:56.960,0:01:57.230
Right?

0:01:57.230,0:01:58.070
You have the same base.

0:01:58.070,0:01:59.840
When you multiply,[br]you can add exponents.

0:01:59.840,0:02:02.960
So this would be i[br]to the 501st power.

0:02:02.960,0:02:05.170
And we know that this[br]is the same thing

0:02:05.170,0:02:07.920
as-- i to the 500th power[br]is the same thing as i

0:02:07.920,0:02:10.050
to the fourth power.

0:02:10.050,0:02:11.700
4 times what?

0:02:11.700,0:02:14.760
4 times 125 is 500.

0:02:14.760,0:02:17.280
So that's this part right[br]over here. i to the 500th

0:02:17.280,0:02:21.510
is the same thing as i to the[br]fourth to the 125th power.

0:02:21.510,0:02:26.150
And then that times[br]i to the first power.

0:02:26.150,0:02:27.800
Well, i to the fourth is 1.

0:02:27.800,0:02:31.690
1 to the 125th power[br]is just going to be 1.

0:02:31.690,0:02:33.130
This whole thing is 1.

0:02:33.130,0:02:37.140
And so we are just left[br]with i to the first.

0:02:37.140,0:02:39.222
So this is going[br]to be equal to i.

0:02:39.222,0:02:41.430
So it seems like a really[br]daunting problem, something

0:02:41.430,0:02:43.180
that you would have[br]to sit and do all day,

0:02:43.180,0:02:46.090
but you can use this cycling[br]to realize look, i to the 500th

0:02:46.090,0:02:47.620
is just going to be 1.

0:02:47.620,0:02:51.690
And so i to the 501th is just[br]going to be i times that.

0:02:51.690,0:02:55.060
So i to any multiple of 4--[br]let me write this generally.

0:02:55.060,0:03:00.450
So if you have i to any multiple[br]of 4, so this right over here

0:03:00.450,0:03:04.030
is-- well, we'll just restrict k[br]to be non-negative right now. k

0:03:04.030,0:03:06.380
is greater than or equal to 0.

0:03:06.380,0:03:10.250
So if we have i to any[br]multiple of 4, right over here,

0:03:10.250,0:03:16.130
we are going to get 1, because[br]this is the same thing as i

0:03:16.130,0:03:19.280
to the fourth power[br]to the k-th power.

0:03:19.280,0:03:22.180
And that is the same thing[br]as 1 to the k-th power,

0:03:22.180,0:03:23.960
which is clearly equal to 1.

0:03:23.960,0:03:25.510
And if we have[br]anything else-- if we

0:03:25.510,0:03:29.340
have i to the 4k plus 1 power,[br]i to the 4k plus 2 power,

0:03:29.340,0:03:31.640
we can then just do this[br]technique right over here.

0:03:31.640,0:03:33.640
So let's try that with a[br]few more problems, just

0:03:33.640,0:03:35.920
to make it clear that[br]you can do really,

0:03:35.920,0:03:38.200
really arbitrarily crazy things.

0:03:38.200,0:03:45.020
So let's take i to[br]the 7,321st power.

0:03:45.020,0:03:47.540
Now, we just have[br]to figure out this

0:03:47.540,0:03:52.940
is going to be some multiple[br]of 4 plus something else.

0:03:52.940,0:03:55.870
So to do that, well, you could[br]just look at it by sight,

0:03:55.870,0:03:58.870
that 7,320 is divisible by 4.

0:03:58.870,0:04:00.270
You can verify that by hand.

0:04:00.270,0:04:02.160
And then you have[br]that 1 left over.

0:04:02.160,0:04:08.020
And so this is going to[br]be i to the 7,320 times

0:04:08.020,0:04:09.770
i to the first power.

0:04:09.770,0:04:12.905
This is a multiple of 4-- this[br]right here is a multiple of 4--

0:04:12.905,0:04:17.240
and I know that because[br]any 1,000 is multiple of 4,

0:04:17.240,0:04:21.209
any 100 is a multiple of 4,[br]and then 20 is a multiple of 4.

0:04:21.209,0:04:24.497
And so this right over[br]here will simplify to 1.

0:04:24.497,0:04:26.080
Sorry, that's not i[br]to the i-th power.

0:04:26.080,0:04:28.960
This is i to the first power.

0:04:28.960,0:04:33.240
7,321 is 7,320 plus 1.

0:04:33.240,0:04:37.287
And so this part right over[br]here is going to simplify to 1,

0:04:37.287,0:04:38.870
and we're just going[br]to be left with i

0:04:38.870,0:04:41.100
to the first power, or just i.

0:04:41.100,0:04:42.600
Let's do another one.

0:04:42.600,0:04:50.860
i to the 90-- let me try[br]something interesting.

0:04:54.030,0:04:56.230
i to the 99th power.

0:04:56.230,0:04:58.860
So once again, what's[br]the highest multiple

0:04:58.860,0:05:01.490
of 4 that is less than 99?

0:05:01.490,0:05:02.590
It is 96.

0:05:05.230,0:05:08.930
So this is the same thing[br]as i to the 96th power times

0:05:08.930,0:05:11.400
i to the third power, right?

0:05:11.400,0:05:14.320
If you multiply these, same[br]base, add the exponent,

0:05:14.320,0:05:16.840
you would get i[br]to the 99th power.

0:05:16.840,0:05:20.410
i to the 96th power, since[br]this is a multiple of 4,

0:05:20.410,0:05:23.740
this is i to the fourth, and[br]then that to the 16th power.

0:05:23.740,0:05:26.850
So that's just 1 to the[br]16th, so this is just 1.

0:05:26.850,0:05:29.670
And then you're just left[br]with i to the third power.

0:05:29.670,0:05:32.940
And you could either remember[br]that i to the third power

0:05:32.940,0:05:35.630
is equal to-- you[br]can just remember

0:05:35.630,0:05:36.880
that it's equal to negative i.

0:05:36.880,0:05:39.270
Or if you forget that,[br]you could just say, look,

0:05:39.270,0:05:42.480
this is the same thing[br]as i squared times i.

0:05:42.480,0:05:45.360
This is equal to[br]i squared times i.

0:05:45.360,0:05:48.800
i squared, by definition,[br]is equal to negative 1.

0:05:48.800,0:05:55.340
So you have negative 1 times[br]i is equal to negative i.

0:05:55.340,0:05:58.890
Let me do one more[br]just for the fun of it.

0:05:58.890,0:06:01.840
Let's take i to the 38th power.

0:06:01.840,0:06:03.450
Well, once again,[br]this is equal to i

0:06:03.450,0:06:07.230
to the 36th times i squared.

0:06:07.230,0:06:09.040
I'm doing i to the 36th[br]power, since that's

0:06:09.040,0:06:11.920
the largest multiple[br]of 4 that goes into 38.

0:06:11.920,0:06:13.730
What's left over is this 2.

0:06:13.730,0:06:15.870
This simplifies[br]to 1, and I'm just

0:06:15.870,0:06:20.530
left with i squared, which[br]is equal to negative 1.