[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.60,0:00:04.98,Default,,0000,0000,0000,,I don't think we can do enough\Nvideos on why raising something
Dialogue: 0,0:00:04.98,0:00:09.05,Default,,0000,0000,0000,,to a negative exponent\Nis equivalent to 1
Dialogue: 0,0:00:09.05,0:00:17.39,Default,,0000,0000,0000,,over that base raised to\Nthe positive exponent,
Dialogue: 0,0:00:17.39,0:00:18.31,Default,,0000,0000,0000,,I should say.
Dialogue: 0,0:00:18.31,0:00:22.20,Default,,0000,0000,0000,,And to get more intuition\Nabout why this makes sense,
Dialogue: 0,0:00:22.20,0:00:24.94,Default,,0000,0000,0000,,I look at different\Npowers of 2, and then
Dialogue: 0,0:00:24.94,0:00:29.10,Default,,0000,0000,0000,,think about what makes sense\Nas we go to exponents below 0,
Dialogue: 0,0:00:29.10,0:00:32.15,Default,,0000,0000,0000,,integer exponents below 0.
Dialogue: 0,0:00:32.15,0:00:37.47,Default,,0000,0000,0000,,So let's start with\N2 to the third power.
Dialogue: 0,0:00:37.47,0:00:39.12,Default,,0000,0000,0000,,Well, 2 to the third\Npower is 2 times
Dialogue: 0,0:00:39.12,0:00:42.25,Default,,0000,0000,0000,,2 times 2, which of\Ncourse is equal to 8.
Dialogue: 0,0:00:42.25,0:00:45.12,Default,,0000,0000,0000,,Now what about 2 to\Nthe second power?
Dialogue: 0,0:00:45.12,0:00:47.10,Default,,0000,0000,0000,,Well that's going to\Nbe 2 times 2, which
Dialogue: 0,0:00:47.10,0:00:49.24,Default,,0000,0000,0000,,is of course equal to 4.
Dialogue: 0,0:00:49.24,0:00:50.95,Default,,0000,0000,0000,,And to go from 2\Nto the third to 2
Dialogue: 0,0:00:50.95,0:00:52.85,Default,,0000,0000,0000,,to the second power,\Nwhat happened here?
Dialogue: 0,0:00:52.85,0:00:54.32,Default,,0000,0000,0000,,Well, we divided by 2.
Dialogue: 0,0:00:56.86,0:00:58.06,Default,,0000,0000,0000,,Now, let's keep going.
Dialogue: 0,0:00:58.06,0:01:00.57,Default,,0000,0000,0000,,What about 2 to the first power?
Dialogue: 0,0:01:00.57,0:01:02.57,Default,,0000,0000,0000,,Well, that's just\N2, and once again,
Dialogue: 0,0:01:02.57,0:01:08.73,Default,,0000,0000,0000,,to go from 2 squared to 2 to the\Nfirst power, we divided by 2.
Dialogue: 0,0:01:08.73,0:01:10.98,Default,,0000,0000,0000,,Now things are going\Nto get interesting.
Dialogue: 0,0:01:10.98,0:01:13.47,Default,,0000,0000,0000,,2 to the 0-th power,\Nand actually this
Dialogue: 0,0:01:13.47,0:01:16.30,Default,,0000,0000,0000,,will help to build the\Nintuition of why it's something
Dialogue: 0,0:01:16.30,0:01:19.71,Default,,0000,0000,0000,,nonzero to the 0-th\Npower is defined to be 1.
Dialogue: 0,0:01:19.71,0:01:23.97,Default,,0000,0000,0000,,Well, so far, every time we\Ndecremented the exponent by 1,
Dialogue: 0,0:01:23.97,0:01:28.63,Default,,0000,0000,0000,,we essentially divided by 2,\Nso we should divide by 2 again.
Dialogue: 0,0:01:28.63,0:01:31.56,Default,,0000,0000,0000,,So if we divide by\N2 again, we get 1.
Dialogue: 0,0:01:31.56,0:01:32.98,Default,,0000,0000,0000,,And this is part\Nof the motivation
Dialogue: 0,0:01:32.98,0:01:36.73,Default,,0000,0000,0000,,of why 2 to the 0 power\Nshould be equal to 1.
Dialogue: 0,0:01:36.73,0:01:38.74,Default,,0000,0000,0000,,But let's keep going.
Dialogue: 0,0:01:38.74,0:01:41.60,Default,,0000,0000,0000,,What should 2 to\Nthe negative 1 power
Dialogue: 0,0:01:41.60,0:01:43.86,Default,,0000,0000,0000,,be, if we want to be\Nconsistent about continuing
Dialogue: 0,0:01:43.86,0:01:45.53,Default,,0000,0000,0000,,to divide by 2?
Dialogue: 0,0:01:45.53,0:01:49.64,Default,,0000,0000,0000,,Well, we divide by\N2 again, and so this
Dialogue: 0,0:01:49.64,0:01:52.32,Default,,0000,0000,0000,,is going to be equal to 1/2.
Dialogue: 0,0:01:52.32,0:01:54.69,Default,,0000,0000,0000,,Notice, 2 to the\Nnegative 1 is 1/2.
Dialogue: 0,0:01:54.69,0:01:57.04,Default,,0000,0000,0000,,2 to the 1 is equal to 1.
Dialogue: 0,0:01:57.04,0:02:00.48,Default,,0000,0000,0000,,This is equal to the\Nreciprocal of this.
Dialogue: 0,0:02:00.48,0:02:01.32,Default,,0000,0000,0000,,Let's keep going.
Dialogue: 0,0:02:01.32,0:02:02.94,Default,,0000,0000,0000,,This is fun.
Dialogue: 0,0:02:02.94,0:02:06.29,Default,,0000,0000,0000,,So what should 2 to the\Nnegative 2 power be?
Dialogue: 0,0:02:06.29,0:02:08.95,Default,,0000,0000,0000,,Well, we should\Ndivide by 2 again.
Dialogue: 0,0:02:08.95,0:02:12.13,Default,,0000,0000,0000,,Divide by 2 again,\Nyou get to 1/4.
Dialogue: 0,0:02:12.13,0:02:13.53,Default,,0000,0000,0000,,I think you see the pattern.
Dialogue: 0,0:02:13.53,0:02:18.08,Default,,0000,0000,0000,,2 to the negative 3 power, well\Nwe should divide by 2 again.
Dialogue: 0,0:02:18.08,0:02:23.37,Default,,0000,0000,0000,,And we get to 1/8, which\Nis the reciprocal of 2
Dialogue: 0,0:02:23.37,0:02:25.11,Default,,0000,0000,0000,,to the third power.
Dialogue: 0,0:02:25.11,0:02:27.17,Default,,0000,0000,0000,,So once again,\Nanother way to think
Dialogue: 0,0:02:27.17,0:02:33.01,Default,,0000,0000,0000,,about why negative exponents\Nare about taking reciprocals.
Dialogue: 0,0:02:33.01,0:02:36.73,Default,,0000,0000,0000,,Taking something to the negative\Nexponent is equivalent to 1
Dialogue: 0,0:02:36.73,0:02:41.06,Default,,0000,0000,0000,,over taking that same base\Nto the positive exponent.