[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.52,0:00:02.81,Default,,0000,0000,0000,,Now that we've seen\Nthat as we take i
Dialogue: 0,0:00:02.81,0:00:06.74,Default,,0000,0000,0000,,to higher and higher powers,\Nit cycles between 1, i,
Dialogue: 0,0:00:06.74,0:00:11.26,Default,,0000,0000,0000,,negative 1, negative i, then\Nback to 1, i, negative 1,
Dialogue: 0,0:00:11.26,0:00:12.29,Default,,0000,0000,0000,,and negative i.
Dialogue: 0,0:00:12.29,0:00:14.16,Default,,0000,0000,0000,,I want to see if we can\Ntackle some, I guess
Dialogue: 0,0:00:14.16,0:00:15.78,Default,,0000,0000,0000,,you could call them,\Ntrickier problems.
Dialogue: 0,0:00:15.78,0:00:17.11,Default,,0000,0000,0000,,And you might see these surface.
Dialogue: 0,0:00:17.11,0:00:18.53,Default,,0000,0000,0000,,And they're also\Nkind of fun to do
Dialogue: 0,0:00:18.53,0:00:22.18,Default,,0000,0000,0000,,to realize that you can use\Nthe fact that the powers of i
Dialogue: 0,0:00:22.18,0:00:23.32,Default,,0000,0000,0000,,cycle through these values.
Dialogue: 0,0:00:23.32,0:00:25.99,Default,,0000,0000,0000,,You can use this to really,\Non a back of an envelope,
Dialogue: 0,0:00:25.99,0:00:29.10,Default,,0000,0000,0000,,take arbitrarily\Nhigh powers of i.
Dialogue: 0,0:00:29.10,0:00:31.65,Default,,0000,0000,0000,,So let's try, just\Nfor fun, let's
Dialogue: 0,0:00:31.65,0:00:35.31,Default,,0000,0000,0000,,see what i to the\N100th power is.
Dialogue: 0,0:00:35.31,0:00:39.28,Default,,0000,0000,0000,,And the realization here is\Nthat 100 is a multiple of 4.
Dialogue: 0,0:00:39.28,0:00:43.80,Default,,0000,0000,0000,,So you could say that this\Nis the same thing as i
Dialogue: 0,0:00:43.80,0:00:47.47,Default,,0000,0000,0000,,to the 4 times 25th power.
Dialogue: 0,0:00:47.47,0:00:50.05,Default,,0000,0000,0000,,And this is the same thing, just\Nfrom our exponent properties,
Dialogue: 0,0:00:50.05,0:00:55.17,Default,,0000,0000,0000,,as i to the fourth power\Nraised to the 25th power.
Dialogue: 0,0:00:55.17,0:00:57.00,Default,,0000,0000,0000,,If you have something\Nraised to an exponent,
Dialogue: 0,0:00:57.00,0:00:59.09,Default,,0000,0000,0000,,and then that is\Nraised to an exponent,
Dialogue: 0,0:00:59.09,0:01:02.30,Default,,0000,0000,0000,,that's the same thing as\Nmultiplying the two exponents.
Dialogue: 0,0:01:02.30,0:01:04.17,Default,,0000,0000,0000,,And we know that\Ni to the fourth,
Dialogue: 0,0:01:04.17,0:01:05.42,Default,,0000,0000,0000,,that's pretty straightforward.
Dialogue: 0,0:01:05.42,0:01:07.39,Default,,0000,0000,0000,,i to the fourth is just 1.
Dialogue: 0,0:01:07.39,0:01:09.59,Default,,0000,0000,0000,,i to the fourth is\N1, so this is 1.
Dialogue: 0,0:01:09.59,0:01:12.30,Default,,0000,0000,0000,,So this is equal to\N1 to the 25th power,
Dialogue: 0,0:01:12.30,0:01:15.91,Default,,0000,0000,0000,,which is just equal to 1.
Dialogue: 0,0:01:15.91,0:01:18.87,Default,,0000,0000,0000,,So once again, we use this\Nkind of cycling ability of i
Dialogue: 0,0:01:18.87,0:01:20.45,Default,,0000,0000,0000,,when you take its\Npowers to figure out
Dialogue: 0,0:01:20.45,0:01:22.67,Default,,0000,0000,0000,,a very high exponent of i.
Dialogue: 0,0:01:22.67,0:01:24.88,Default,,0000,0000,0000,,Now let's say we try something\Na little bit stranger.
Dialogue: 0,0:01:27.73,0:01:31.20,Default,,0000,0000,0000,,Let's try i to the 501st power.
Dialogue: 0,0:01:31.20,0:01:34.62,Default,,0000,0000,0000,,Now in this situation, 501,\Nit's not a multiple of 4.
Dialogue: 0,0:01:34.62,0:01:36.31,Default,,0000,0000,0000,,So you can't just\Ndo that that simply.
Dialogue: 0,0:01:36.31,0:01:38.23,Default,,0000,0000,0000,,But what you could do,\Nis you could write this
Dialogue: 0,0:01:38.23,0:01:41.50,Default,,0000,0000,0000,,as a product of two\Nnumbers, one that
Dialogue: 0,0:01:41.50,0:01:44.14,Default,,0000,0000,0000,,is i to a multiple\Nof fourth power.
Dialogue: 0,0:01:44.14,0:01:45.58,Default,,0000,0000,0000,,And then one that isn't.
Dialogue: 0,0:01:45.58,0:01:47.05,Default,,0000,0000,0000,,And so you could rewrite this.
Dialogue: 0,0:01:47.05,0:01:50.39,Default,,0000,0000,0000,,500 is a multiple of 4.
Dialogue: 0,0:01:50.39,0:01:56.00,Default,,0000,0000,0000,,So you could write this as\Ni to the 500th power times i
Dialogue: 0,0:01:56.00,0:01:56.96,Default,,0000,0000,0000,,to the first power.
Dialogue: 0,0:01:56.96,0:01:57.23,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:01:57.23,0:01:58.07,Default,,0000,0000,0000,,You have the same base.
Dialogue: 0,0:01:58.07,0:01:59.84,Default,,0000,0000,0000,,When you multiply,\Nyou can add exponents.
Dialogue: 0,0:01:59.84,0:02:02.96,Default,,0000,0000,0000,,So this would be i\Nto the 501st power.
Dialogue: 0,0:02:02.96,0:02:05.17,Default,,0000,0000,0000,,And we know that this\Nis the same thing
Dialogue: 0,0:02:05.17,0:02:07.92,Default,,0000,0000,0000,,as-- i to the 500th power\Nis the same thing as i
Dialogue: 0,0:02:07.92,0:02:10.05,Default,,0000,0000,0000,,to the fourth power.
Dialogue: 0,0:02:10.05,0:02:11.70,Default,,0000,0000,0000,,4 times what?
Dialogue: 0,0:02:11.70,0:02:14.76,Default,,0000,0000,0000,,4 times 125 is 500.
Dialogue: 0,0:02:14.76,0:02:17.28,Default,,0000,0000,0000,,So that's this part right\Nover here. i to the 500th
Dialogue: 0,0:02:17.28,0:02:21.51,Default,,0000,0000,0000,,is the same thing as i to the\Nfourth to the 125th power.
Dialogue: 0,0:02:21.51,0:02:26.15,Default,,0000,0000,0000,,And then that times\Ni to the first power.
Dialogue: 0,0:02:26.15,0:02:27.80,Default,,0000,0000,0000,,Well, i to the fourth is 1.
Dialogue: 0,0:02:27.80,0:02:31.69,Default,,0000,0000,0000,,1 to the 125th power\Nis just going to be 1.
Dialogue: 0,0:02:31.69,0:02:33.13,Default,,0000,0000,0000,,This whole thing is 1.
Dialogue: 0,0:02:33.13,0:02:37.14,Default,,0000,0000,0000,,And so we are just left\Nwith i to the first.
Dialogue: 0,0:02:37.14,0:02:39.22,Default,,0000,0000,0000,,So this is going\Nto be equal to i.
Dialogue: 0,0:02:39.22,0:02:41.43,Default,,0000,0000,0000,,So it seems like a really\Ndaunting problem, something
Dialogue: 0,0:02:41.43,0:02:43.18,Default,,0000,0000,0000,,that you would have\Nto sit and do all day,
Dialogue: 0,0:02:43.18,0:02:46.09,Default,,0000,0000,0000,,but you can use this cycling\Nto realize look, i to the 500th
Dialogue: 0,0:02:46.09,0:02:47.62,Default,,0000,0000,0000,,is just going to be 1.
Dialogue: 0,0:02:47.62,0:02:51.69,Default,,0000,0000,0000,,And so i to the 501th is just\Ngoing to be i times that.
Dialogue: 0,0:02:51.69,0:02:55.06,Default,,0000,0000,0000,,So i to any multiple of 4--\Nlet me write this generally.
Dialogue: 0,0:02:55.06,0:03:00.45,Default,,0000,0000,0000,,So if you have i to any multiple\Nof 4, so this right over here
Dialogue: 0,0:03:00.45,0:03:04.03,Default,,0000,0000,0000,,is-- well, we'll just restrict k\Nto be non-negative right now. k
Dialogue: 0,0:03:04.03,0:03:06.38,Default,,0000,0000,0000,,is greater than or equal to 0.
Dialogue: 0,0:03:06.38,0:03:10.25,Default,,0000,0000,0000,,So if we have i to any\Nmultiple of 4, right over here,
Dialogue: 0,0:03:10.25,0:03:16.13,Default,,0000,0000,0000,,we are going to get 1, because\Nthis is the same thing as i
Dialogue: 0,0:03:16.13,0:03:19.28,Default,,0000,0000,0000,,to the fourth power\Nto the k-th power.
Dialogue: 0,0:03:19.28,0:03:22.18,Default,,0000,0000,0000,,And that is the same thing\Nas 1 to the k-th power,
Dialogue: 0,0:03:22.18,0:03:23.96,Default,,0000,0000,0000,,which is clearly equal to 1.
Dialogue: 0,0:03:23.96,0:03:25.51,Default,,0000,0000,0000,,And if we have\Nanything else-- if we
Dialogue: 0,0:03:25.51,0:03:29.34,Default,,0000,0000,0000,,have i to the 4k plus 1 power,\Ni to the 4k plus 2 power,
Dialogue: 0,0:03:29.34,0:03:31.64,Default,,0000,0000,0000,,we can then just do this\Ntechnique right over here.
Dialogue: 0,0:03:31.64,0:03:33.64,Default,,0000,0000,0000,,So let's try that with a\Nfew more problems, just
Dialogue: 0,0:03:33.64,0:03:35.92,Default,,0000,0000,0000,,to make it clear that\Nyou can do really,
Dialogue: 0,0:03:35.92,0:03:38.20,Default,,0000,0000,0000,,really arbitrarily crazy things.
Dialogue: 0,0:03:38.20,0:03:45.02,Default,,0000,0000,0000,,So let's take i to\Nthe 7,321st power.
Dialogue: 0,0:03:45.02,0:03:47.54,Default,,0000,0000,0000,,Now, we just have\Nto figure out this
Dialogue: 0,0:03:47.54,0:03:52.94,Default,,0000,0000,0000,,is going to be some multiple\Nof 4 plus something else.
Dialogue: 0,0:03:52.94,0:03:55.87,Default,,0000,0000,0000,,So to do that, well, you could\Njust look at it by sight,
Dialogue: 0,0:03:55.87,0:03:58.87,Default,,0000,0000,0000,,that 7,320 is divisible by 4.
Dialogue: 0,0:03:58.87,0:04:00.27,Default,,0000,0000,0000,,You can verify that by hand.
Dialogue: 0,0:04:00.27,0:04:02.16,Default,,0000,0000,0000,,And then you have\Nthat 1 left over.
Dialogue: 0,0:04:02.16,0:04:08.02,Default,,0000,0000,0000,,And so this is going to\Nbe i to the 7,320 times
Dialogue: 0,0:04:08.02,0:04:09.77,Default,,0000,0000,0000,,i to the first power.
Dialogue: 0,0:04:09.77,0:04:12.90,Default,,0000,0000,0000,,This is a multiple of 4-- this\Nright here is a multiple of 4--
Dialogue: 0,0:04:12.90,0:04:17.24,Default,,0000,0000,0000,,and I know that because\Nany 1,000 is multiple of 4,
Dialogue: 0,0:04:17.24,0:04:21.21,Default,,0000,0000,0000,,any 100 is a multiple of 4,\Nand then 20 is a multiple of 4.
Dialogue: 0,0:04:21.21,0:04:24.50,Default,,0000,0000,0000,,And so this right over\Nhere will simplify to 1.
Dialogue: 0,0:04:24.50,0:04:26.08,Default,,0000,0000,0000,,Sorry, that's not i\Nto the i-th power.
Dialogue: 0,0:04:26.08,0:04:28.96,Default,,0000,0000,0000,,This is i to the first power.
Dialogue: 0,0:04:28.96,0:04:33.24,Default,,0000,0000,0000,,7,321 is 7,320 plus 1.
Dialogue: 0,0:04:33.24,0:04:37.29,Default,,0000,0000,0000,,And so this part right over\Nhere is going to simplify to 1,
Dialogue: 0,0:04:37.29,0:04:38.87,Default,,0000,0000,0000,,and we're just going\Nto be left with i
Dialogue: 0,0:04:38.87,0:04:41.10,Default,,0000,0000,0000,,to the first power, or just i.
Dialogue: 0,0:04:41.10,0:04:42.60,Default,,0000,0000,0000,,Let's do another one.
Dialogue: 0,0:04:42.60,0:04:50.86,Default,,0000,0000,0000,,i to the 90-- let me try\Nsomething interesting.
Dialogue: 0,0:04:54.03,0:04:56.23,Default,,0000,0000,0000,,i to the 99th power.
Dialogue: 0,0:04:56.23,0:04:58.86,Default,,0000,0000,0000,,So once again, what's\Nthe highest multiple
Dialogue: 0,0:04:58.86,0:05:01.49,Default,,0000,0000,0000,,of 4 that is less than 99?
Dialogue: 0,0:05:01.49,0:05:02.59,Default,,0000,0000,0000,,It is 96.
Dialogue: 0,0:05:05.23,0:05:08.93,Default,,0000,0000,0000,,So this is the same thing\Nas i to the 96th power times
Dialogue: 0,0:05:08.93,0:05:11.40,Default,,0000,0000,0000,,i to the third power, right?
Dialogue: 0,0:05:11.40,0:05:14.32,Default,,0000,0000,0000,,If you multiply these, same\Nbase, add the exponent,
Dialogue: 0,0:05:14.32,0:05:16.84,Default,,0000,0000,0000,,you would get i\Nto the 99th power.
Dialogue: 0,0:05:16.84,0:05:20.41,Default,,0000,0000,0000,,i to the 96th power, since\Nthis is a multiple of 4,
Dialogue: 0,0:05:20.41,0:05:23.74,Default,,0000,0000,0000,,this is i to the fourth, and\Nthen that to the 16th power.
Dialogue: 0,0:05:23.74,0:05:26.85,Default,,0000,0000,0000,,So that's just 1 to the\N16th, so this is just 1.
Dialogue: 0,0:05:26.85,0:05:29.67,Default,,0000,0000,0000,,And then you're just left\Nwith i to the third power.
Dialogue: 0,0:05:29.67,0:05:32.94,Default,,0000,0000,0000,,And you could either remember\Nthat i to the third power
Dialogue: 0,0:05:32.94,0:05:35.63,Default,,0000,0000,0000,,is equal to-- you\Ncan just remember
Dialogue: 0,0:05:35.63,0:05:36.88,Default,,0000,0000,0000,,that it's equal to negative i.
Dialogue: 0,0:05:36.88,0:05:39.27,Default,,0000,0000,0000,,Or if you forget that,\Nyou could just say, look,
Dialogue: 0,0:05:39.27,0:05:42.48,Default,,0000,0000,0000,,this is the same thing\Nas i squared times i.
Dialogue: 0,0:05:42.48,0:05:45.36,Default,,0000,0000,0000,,This is equal to\Ni squared times i.
Dialogue: 0,0:05:45.36,0:05:48.80,Default,,0000,0000,0000,,i squared, by definition,\Nis equal to negative 1.
Dialogue: 0,0:05:48.80,0:05:55.34,Default,,0000,0000,0000,,So you have negative 1 times\Ni is equal to negative i.
Dialogue: 0,0:05:55.34,0:05:58.89,Default,,0000,0000,0000,,Let me do one more\Njust for the fun of it.
Dialogue: 0,0:05:58.89,0:06:01.84,Default,,0000,0000,0000,,Let's take i to the 38th power.
Dialogue: 0,0:06:01.84,0:06:03.45,Default,,0000,0000,0000,,Well, once again,\Nthis is equal to i
Dialogue: 0,0:06:03.45,0:06:07.23,Default,,0000,0000,0000,,to the 36th times i squared.
Dialogue: 0,0:06:07.23,0:06:09.04,Default,,0000,0000,0000,,I'm doing i to the 36th\Npower, since that's
Dialogue: 0,0:06:09.04,0:06:11.92,Default,,0000,0000,0000,,the largest multiple\Nof 4 that goes into 38.
Dialogue: 0,0:06:11.92,0:06:13.73,Default,,0000,0000,0000,,What's left over is this 2.
Dialogue: 0,0:06:13.73,0:06:15.87,Default,,0000,0000,0000,,This simplifies\Nto 1, and I'm just
Dialogue: 0,0:06:15.87,0:06:20.53,Default,,0000,0000,0000,,left with i squared, which\Nis equal to negative 1.