[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.48,Default,,0000,0000,0000,,I have here a bunch of radical\Nexpressions, or square root
Dialogue: 0,0:00:04.48,0:00:05.11,Default,,0000,0000,0000,,expressions.
Dialogue: 0,0:00:05.11,0:00:07.60,Default,,0000,0000,0000,,And what I'm going to do is\Ngo through all of them and
Dialogue: 0,0:00:07.60,0:00:08.50,Default,,0000,0000,0000,,simplify them.
Dialogue: 0,0:00:08.50,0:00:11.24,Default,,0000,0000,0000,,And we'll talk about whether\Nthese are rational or
Dialogue: 0,0:00:11.24,0:00:13.39,Default,,0000,0000,0000,,irrational numbers.
Dialogue: 0,0:00:13.39,0:00:15.71,Default,,0000,0000,0000,,So let's start with A.
Dialogue: 0,0:00:15.71,0:00:20.44,Default,,0000,0000,0000,,A is equal to the square\Nroot of 25.
Dialogue: 0,0:00:20.44,0:00:26.56,Default,,0000,0000,0000,,Well that's the same thing as\Nthe square root of 5 times 5,
Dialogue: 0,0:00:26.56,0:00:31.00,Default,,0000,0000,0000,,which is a clearly\Ngoing to be 5.
Dialogue: 0,0:00:31.00,0:00:34.44,Default,,0000,0000,0000,,We're focusing on the positive\Nsquare root here.
Dialogue: 0,0:00:34.44,0:00:37.06,Default,,0000,0000,0000,,Now let's do B.
Dialogue: 0,0:00:37.06,0:00:39.92,Default,,0000,0000,0000,,B I'll do in a different color,\Nfor the principal root,
Dialogue: 0,0:00:39.92,0:00:42.25,Default,,0000,0000,0000,,when we say positive\Nsquare root.
Dialogue: 0,0:00:42.25,0:00:46.20,Default,,0000,0000,0000,,B, we have the square\Nroot of 24.
Dialogue: 0,0:00:46.20,0:00:47.96,Default,,0000,0000,0000,,So what you want to do, is\Nyou want to get the prime
Dialogue: 0,0:00:47.96,0:00:50.53,Default,,0000,0000,0000,,factorization of this\Nnumber right here.
Dialogue: 0,0:00:50.53,0:00:53.56,Default,,0000,0000,0000,,So 24, let's do its prime\Nfactorization.
Dialogue: 0,0:00:53.56,0:00:56.25,Default,,0000,0000,0000,,This is 2 times 12.
Dialogue: 0,0:00:56.25,0:00:59.72,Default,,0000,0000,0000,,12 is 2 times 6.
Dialogue: 0,0:00:59.72,0:01:03.43,Default,,0000,0000,0000,,6 is 2 times 3.
Dialogue: 0,0:01:03.43,0:01:07.22,Default,,0000,0000,0000,,So the square root of 24, this\Nis the same thing as the
Dialogue: 0,0:01:07.22,0:01:15.32,Default,,0000,0000,0000,,square root of 2 times\N2 times 2 times 3.
Dialogue: 0,0:01:15.32,0:01:18.08,Default,,0000,0000,0000,,That's the same thing as 24.
Dialogue: 0,0:01:18.08,0:01:22.53,Default,,0000,0000,0000,,Well, we see here, we have one\Nperfect square right there.
Dialogue: 0,0:01:22.53,0:01:23.87,Default,,0000,0000,0000,,So we could rewrite this.
Dialogue: 0,0:01:23.87,0:01:30.33,Default,,0000,0000,0000,,This is the same thing as the\Nsquare root of 2 times 2 times
Dialogue: 0,0:01:30.33,0:01:34.03,Default,,0000,0000,0000,,the square root of 2 times 3.
Dialogue: 0,0:01:34.03,0:01:35.89,Default,,0000,0000,0000,,Now this is clearly 2.
Dialogue: 0,0:01:35.89,0:01:37.01,Default,,0000,0000,0000,,This is the square root of 4.
Dialogue: 0,0:01:37.01,0:01:38.92,Default,,0000,0000,0000,,The square root of 4 is 2.
Dialogue: 0,0:01:38.92,0:01:40.71,Default,,0000,0000,0000,,And then this we can't\Nsimplify anymore.
Dialogue: 0,0:01:40.71,0:01:44.52,Default,,0000,0000,0000,,We don't see two numbers\Nmultiplied by itself here.
Dialogue: 0,0:01:44.52,0:01:47.94,Default,,0000,0000,0000,,So this is going to be times\Nthe square root of 6.
Dialogue: 0,0:01:47.94,0:01:50.11,Default,,0000,0000,0000,,Or we could even right this as\Nthe square root of 2 times the
Dialogue: 0,0:01:50.11,0:01:51.54,Default,,0000,0000,0000,,square root of 3.
Dialogue: 0,0:01:51.54,0:01:53.21,Default,,0000,0000,0000,,Now I said I would talk\Nabout whether things
Dialogue: 0,0:01:53.21,0:01:54.55,Default,,0000,0000,0000,,are rational or not.
Dialogue: 0,0:01:54.55,0:01:56.46,Default,,0000,0000,0000,,This is rational.
Dialogue: 0,0:01:56.46,0:02:03.63,Default,,0000,0000,0000,,This part A can be expressed\Nas the ratio of 2 integers.
Dialogue: 0,0:02:03.63,0:02:05.92,Default,,0000,0000,0000,,Namely 5/1.
Dialogue: 0,0:02:05.92,0:02:07.34,Default,,0000,0000,0000,,This is rational.
Dialogue: 0,0:02:07.34,0:02:08.59,Default,,0000,0000,0000,,This is irrational.
Dialogue: 0,0:02:11.84,0:02:14.06,Default,,0000,0000,0000,,I'm not going to prove\Nit in this video.
Dialogue: 0,0:02:14.06,0:02:18.77,Default,,0000,0000,0000,,But anything that is the product\Nof irrational numbers.
Dialogue: 0,0:02:18.77,0:02:24.92,Default,,0000,0000,0000,,And the square root of any prime\Nnumber is irrational.
Dialogue: 0,0:02:24.92,0:02:25.79,Default,,0000,0000,0000,,I'm not proving it here.
Dialogue: 0,0:02:25.79,0:02:29.06,Default,,0000,0000,0000,,This is the square root of 2\Ntimes the square root of 3.
Dialogue: 0,0:02:29.06,0:02:30.36,Default,,0000,0000,0000,,That's what the square\Nroot of 6 is.
Dialogue: 0,0:02:30.36,0:02:32.28,Default,,0000,0000,0000,,And that's what makes\Nthis irrational.
Dialogue: 0,0:02:32.28,0:02:35.91,Default,,0000,0000,0000,,I cannot express this as\Nany type of fraction.
Dialogue: 0,0:02:35.91,0:02:40.83,Default,,0000,0000,0000,,I can't express this as some\Ninteger over some other
Dialogue: 0,0:02:40.83,0:02:42.28,Default,,0000,0000,0000,,integer like I did there.
Dialogue: 0,0:02:42.28,0:02:43.25,Default,,0000,0000,0000,,And I'm not proving it here.
Dialogue: 0,0:02:43.25,0:02:45.91,Default,,0000,0000,0000,,I'm just giving you a little\Nbit of practice.
Dialogue: 0,0:02:45.91,0:02:47.01,Default,,0000,0000,0000,,And a quicker way to do this.
Dialogue: 0,0:02:47.01,0:02:48.30,Default,,0000,0000,0000,,You could say, hey,\N4 goes into this.
Dialogue: 0,0:02:48.30,0:02:49.77,Default,,0000,0000,0000,,4 is a perfect square.
Dialogue: 0,0:02:49.77,0:02:50.83,Default,,0000,0000,0000,,Let me take a 4 out.
Dialogue: 0,0:02:50.83,0:02:52.12,Default,,0000,0000,0000,,This is 4 times 6.
Dialogue: 0,0:02:52.12,0:02:54.77,Default,,0000,0000,0000,,The square root of 4 is 2, leave\Nthe 6 in, and you would
Dialogue: 0,0:02:54.77,0:02:56.16,Default,,0000,0000,0000,,have gotten the 2 square\Nroots of 6.
Dialogue: 0,0:02:56.16,0:02:58.99,Default,,0000,0000,0000,,Which you will get the hang of\Nit eventually, but I want to
Dialogue: 0,0:02:58.99,0:03:01.59,Default,,0000,0000,0000,,do it systematically first.
Dialogue: 0,0:03:01.59,0:03:03.82,Default,,0000,0000,0000,,Let's do part C.
Dialogue: 0,0:03:03.82,0:03:06.61,Default,,0000,0000,0000,,Square root of 20.
Dialogue: 0,0:03:06.61,0:03:12.35,Default,,0000,0000,0000,,Once again, 20 is 2 times\N10, which is 2 times 5.
Dialogue: 0,0:03:12.35,0:03:18.05,Default,,0000,0000,0000,,So this is the same thing as the\Nsquare root of 2 times 2,
Dialogue: 0,0:03:18.05,0:03:20.74,Default,,0000,0000,0000,,right, times 5.
Dialogue: 0,0:03:20.74,0:03:22.69,Default,,0000,0000,0000,,Now, the square root of 2 times\N2, that's clearly just
Dialogue: 0,0:03:22.69,0:03:25.12,Default,,0000,0000,0000,,going to be 2.
Dialogue: 0,0:03:25.12,0:03:26.53,Default,,0000,0000,0000,,It's going to be the square\Nroot of this times
Dialogue: 0,0:03:26.53,0:03:27.38,Default,,0000,0000,0000,,square root of that.
Dialogue: 0,0:03:27.38,0:03:29.40,Default,,0000,0000,0000,,2 times the square root of 5.
Dialogue: 0,0:03:29.40,0:03:31.09,Default,,0000,0000,0000,,And once again, you could\Nprobably do that in your head
Dialogue: 0,0:03:31.09,0:03:31.91,Default,,0000,0000,0000,,with a little practice.
Dialogue: 0,0:03:31.91,0:03:34.92,Default,,0000,0000,0000,,The square root of the\N20 is 4 times 5.
Dialogue: 0,0:03:34.92,0:03:36.55,Default,,0000,0000,0000,,The square root of 4 is 2.
Dialogue: 0,0:03:36.55,0:03:39.08,Default,,0000,0000,0000,,You leave the 5 in\Nthe radical.
Dialogue: 0,0:03:39.08,0:03:43.20,Default,,0000,0000,0000,,So let's do part D.
Dialogue: 0,0:03:43.20,0:03:47.38,Default,,0000,0000,0000,,We have to do the square\Nroot of 200.
Dialogue: 0,0:03:47.38,0:03:48.35,Default,,0000,0000,0000,,Same process.
Dialogue: 0,0:03:48.35,0:03:50.39,Default,,0000,0000,0000,,Let's take the prime\Nfactors of it.
Dialogue: 0,0:03:50.39,0:03:56.31,Default,,0000,0000,0000,,So it's 2 times 100, which is\N2 times 50, which is 2 times
Dialogue: 0,0:03:56.31,0:04:01.03,Default,,0000,0000,0000,,25, which is 5 times 5.
Dialogue: 0,0:04:01.03,0:04:03.64,Default,,0000,0000,0000,,So this right here,\Nwe can rewrite it.
Dialogue: 0,0:04:03.64,0:04:05.80,Default,,0000,0000,0000,,Let me scroll to the\Nright a little bit.
Dialogue: 0,0:04:05.80,0:04:15.03,Default,,0000,0000,0000,,This is equal to the square\Nroot of 2 times 2 times 2
Dialogue: 0,0:04:15.03,0:04:18.39,Default,,0000,0000,0000,,times 5 times 5.
Dialogue: 0,0:04:18.39,0:04:20.73,Default,,0000,0000,0000,,Well we have one perfect square\Nthere, and we have
Dialogue: 0,0:04:20.73,0:04:23.35,Default,,0000,0000,0000,,another perfect square there.
Dialogue: 0,0:04:23.35,0:04:25.29,Default,,0000,0000,0000,,So if I just want to write out\Nall the steps, this would be
Dialogue: 0,0:04:25.29,0:04:31.17,Default,,0000,0000,0000,,the square root of 2 times 2\Ntimes the square root of 2
Dialogue: 0,0:04:31.17,0:04:35.12,Default,,0000,0000,0000,,times the square root\Nof 5 times 5.
Dialogue: 0,0:04:35.12,0:04:37.34,Default,,0000,0000,0000,,The square root of\N2 times 2 is 2.
Dialogue: 0,0:04:37.34,0:04:40.24,Default,,0000,0000,0000,,The square root of 2 is just\Nthe square root of 2.
Dialogue: 0,0:04:40.24,0:04:43.68,Default,,0000,0000,0000,,The square root of 5 times 5,\Nthat's the square root of 25,
Dialogue: 0,0:04:43.68,0:04:45.43,Default,,0000,0000,0000,,that's just going to be 5.
Dialogue: 0,0:04:45.43,0:04:46.88,Default,,0000,0000,0000,,So you can rearrange these.
Dialogue: 0,0:04:46.88,0:04:48.83,Default,,0000,0000,0000,,2 times 5 is 10.
Dialogue: 0,0:04:48.83,0:04:50.73,Default,,0000,0000,0000,,10 square roots of 2.
Dialogue: 0,0:04:50.73,0:04:53.15,Default,,0000,0000,0000,,And once again, this\Nis irrational.
Dialogue: 0,0:04:53.15,0:04:58.80,Default,,0000,0000,0000,,You can't express it as a\Nfraction with an integer and a
Dialogue: 0,0:04:58.80,0:05:00.85,Default,,0000,0000,0000,,numerator and the denominator.
Dialogue: 0,0:05:00.85,0:05:04.27,Default,,0000,0000,0000,,And if you were to actually try\Nto express this number, it
Dialogue: 0,0:05:04.27,0:05:08.61,Default,,0000,0000,0000,,will just keep going on and on\Nand on, and never repeating.
Dialogue: 0,0:05:08.61,0:05:10.79,Default,,0000,0000,0000,,Well let's do part E.
Dialogue: 0,0:05:10.79,0:05:13.72,Default,,0000,0000,0000,,The square root of 2000.
Dialogue: 0,0:05:13.72,0:05:15.66,Default,,0000,0000,0000,,I'll do it down here.
Dialogue: 0,0:05:15.66,0:05:20.62,Default,,0000,0000,0000,,Part E, the square\Nroot of 2000.
Dialogue: 0,0:05:20.62,0:05:23.95,Default,,0000,0000,0000,,Same exact process that we've\Nbeen doing so far.
Dialogue: 0,0:05:23.95,0:05:25.82,Default,,0000,0000,0000,,Let's do the prime\Nfactorization.
Dialogue: 0,0:05:25.82,0:05:35.68,Default,,0000,0000,0000,,That is 2 times 1000, which is\N2 times 500, which is 2 times
Dialogue: 0,0:05:35.68,0:05:45.93,Default,,0000,0000,0000,,250, which is 2 times 125,\Nwhich is 5 times 25,
Dialogue: 0,0:05:45.93,0:05:49.58,Default,,0000,0000,0000,,which is 5 times 5.
Dialogue: 0,0:05:49.58,0:05:50.60,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:05:50.60,0:05:56.18,Default,,0000,0000,0000,,So this is going to be equal to\Nthe square root of 2 times
Dialogue: 0,0:05:56.18,0:05:59.63,Default,,0000,0000,0000,,2-- I'll put it in parentheses--\N2 times 2, times
Dialogue: 0,0:05:59.63,0:06:06.35,Default,,0000,0000,0000,,2 times 2, times 2 times\N2, times 5 times 5,
Dialogue: 0,0:06:06.35,0:06:08.84,Default,,0000,0000,0000,,times 5 times 5, right?
Dialogue: 0,0:06:08.84,0:06:15.39,Default,,0000,0000,0000,,We have 1, 2, 3, 4, 2's, and\Nthen 3, 5's, times 5.
Dialogue: 0,0:06:15.39,0:06:18.00,Default,,0000,0000,0000,,Now what is this going\Nto be equal to?
Dialogue: 0,0:06:18.00,0:06:20.52,Default,,0000,0000,0000,,Well, one thing you might see\Nis, hey, I could write this
Dialogue: 0,0:06:20.52,0:06:25.14,Default,,0000,0000,0000,,as, this is a 4, this is a 4.
Dialogue: 0,0:06:25.14,0:06:27.51,Default,,0000,0000,0000,,So we're going to have\Na 4 repeated.
Dialogue: 0,0:06:27.51,0:06:32.60,Default,,0000,0000,0000,,And so this the same thing as\Nthe square root of 4 times 4
Dialogue: 0,0:06:32.60,0:06:37.33,Default,,0000,0000,0000,,times the square root of\N5 times 5 times the
Dialogue: 0,0:06:37.33,0:06:39.48,Default,,0000,0000,0000,,square root of 5.
Dialogue: 0,0:06:39.48,0:06:42.31,Default,,0000,0000,0000,,So this right here\Nis obviously 4.
Dialogue: 0,0:06:42.31,0:06:44.57,Default,,0000,0000,0000,,This right here is 5.
Dialogue: 0,0:06:44.57,0:06:47.07,Default,,0000,0000,0000,,And then times the\Nsquare root of 5.
Dialogue: 0,0:06:47.07,0:06:52.07,Default,,0000,0000,0000,,So 4 times 5 is 20 square\Nroots of 5.
Dialogue: 0,0:06:52.07,0:06:54.29,Default,,0000,0000,0000,,And once again, this\Nis irrational.
Dialogue: 0,0:06:58.29,0:07:00.99,Default,,0000,0000,0000,,Well, let's do F.
Dialogue: 0,0:07:00.99,0:07:16.85,Default,,0000,0000,0000,,The square root of 1/4, which\Nwe can view this is the same
Dialogue: 0,0:07:16.85,0:07:21.25,Default,,0000,0000,0000,,thing as the square root of 1\Nover the square root of 4,
Dialogue: 0,0:07:21.25,0:07:24.18,Default,,0000,0000,0000,,which is equal to 1/2.
Dialogue: 0,0:07:24.18,0:07:25.17,Default,,0000,0000,0000,,Which is clearly rational.
Dialogue: 0,0:07:25.17,0:07:27.40,Default,,0000,0000,0000,,It can be expressed\Nas a fraction.
Dialogue: 0,0:07:27.40,0:07:33.05,Default,,0000,0000,0000,,So that's clearly rational.
Dialogue: 0,0:07:33.05,0:07:39.38,Default,,0000,0000,0000,,Part G is the square\Nroot of 9/4.
Dialogue: 0,0:07:43.80,0:07:44.60,Default,,0000,0000,0000,,Same logic.
Dialogue: 0,0:07:44.60,0:07:48.16,Default,,0000,0000,0000,,This is equal to the square root\Nof 9 over the square root
Dialogue: 0,0:07:48.16,0:07:52.91,Default,,0000,0000,0000,,of 4, which is equal to 3/2.
Dialogue: 0,0:07:52.91,0:07:56.96,Default,,0000,0000,0000,,Let's do part H.
Dialogue: 0,0:07:56.96,0:08:02.72,Default,,0000,0000,0000,,The square root of 0.16.
Dialogue: 0,0:08:02.72,0:08:05.25,Default,,0000,0000,0000,,Now you could do this in your\Nhead if you immediately
Dialogue: 0,0:08:05.25,0:08:07.67,Default,,0000,0000,0000,,recognize that, gee, if\NI multiply 0.4 times
Dialogue: 0,0:08:07.67,0:08:10.17,Default,,0000,0000,0000,,0.4, I'll get this.
Dialogue: 0,0:08:10.17,0:08:14.19,Default,,0000,0000,0000,,But I'll show you a more\Nsystematic way of doing it, if
Dialogue: 0,0:08:14.19,0:08:16.04,Default,,0000,0000,0000,,that wasn't obvious to you.
Dialogue: 0,0:08:16.04,0:08:18.33,Default,,0000,0000,0000,,So this is the same thing\Nas the square
Dialogue: 0,0:08:18.33,0:08:22.73,Default,,0000,0000,0000,,root of 16/100, right?
Dialogue: 0,0:08:22.73,0:08:24.84,Default,,0000,0000,0000,,That's what 0.16 is.
Dialogue: 0,0:08:24.84,0:08:28.74,Default,,0000,0000,0000,,So this is equal to the square\Nroot of 16 over the square
Dialogue: 0,0:08:28.74,0:08:37.01,Default,,0000,0000,0000,,root of 100, which is equal to\N4/10, which is equal to 0.4.
Dialogue: 0,0:08:37.01,0:08:39.26,Default,,0000,0000,0000,,Let's do a couple\Nmore like that.
Dialogue: 0,0:08:39.26,0:08:39.43,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:08:39.43,0:08:46.18,Default,,0000,0000,0000,,Part I was the square root of\N0.1, which is equal to the
Dialogue: 0,0:08:46.18,0:08:50.84,Default,,0000,0000,0000,,square root of 1/10, which is\Nequal to the square root of 1
Dialogue: 0,0:08:50.84,0:08:55.98,Default,,0000,0000,0000,,over the square root of 10,\Nwhich is equal to 1 over--
Dialogue: 0,0:08:55.98,0:08:59.89,Default,,0000,0000,0000,,now, the square root of 10--\N10 is just 2 times 5.
Dialogue: 0,0:08:59.89,0:09:01.38,Default,,0000,0000,0000,,So that doesn't really\Nhelp us much.
Dialogue: 0,0:09:01.38,0:09:04.92,Default,,0000,0000,0000,,So that's just the square\Nroot of 10 like that.
Dialogue: 0,0:09:04.92,0:09:08.13,Default,,0000,0000,0000,,A lot of math teachers don't\Nlike you leaving that radical
Dialogue: 0,0:09:08.13,0:09:08.87,Default,,0000,0000,0000,,in the denominator.
Dialogue: 0,0:09:08.87,0:09:10.33,Default,,0000,0000,0000,,But I can already tell you\Nthat this is irrational.
Dialogue: 0,0:09:13.94,0:09:15.65,Default,,0000,0000,0000,,You'll just keep getting\Nnumbers.
Dialogue: 0,0:09:15.65,0:09:16.85,Default,,0000,0000,0000,,You can try it on your\Ncalculator, and
Dialogue: 0,0:09:16.85,0:09:17.53,Default,,0000,0000,0000,,it will never repeat.
Dialogue: 0,0:09:17.53,0:09:19.43,Default,,0000,0000,0000,,Your calculator will just give\Nyou an approximation.
Dialogue: 0,0:09:19.43,0:09:21.10,Default,,0000,0000,0000,,Because in order to give the\Nexact value, you'd have to
Dialogue: 0,0:09:21.10,0:09:23.56,Default,,0000,0000,0000,,have an infinite number\Nof digits.
Dialogue: 0,0:09:23.56,0:09:25.77,Default,,0000,0000,0000,,But if you wanted to\Nrationalize this,
Dialogue: 0,0:09:25.77,0:09:26.82,Default,,0000,0000,0000,,just to show you.
Dialogue: 0,0:09:26.82,0:09:28.62,Default,,0000,0000,0000,,If you want to get rid of the\Nradical in the denominator,
Dialogue: 0,0:09:28.62,0:09:32.09,Default,,0000,0000,0000,,you can multiply this times the\Nsquare root of 10 over the
Dialogue: 0,0:09:32.09,0:09:33.52,Default,,0000,0000,0000,,square root of 10, right?
Dialogue: 0,0:09:33.52,0:09:34.91,Default,,0000,0000,0000,,This is just 1.
Dialogue: 0,0:09:34.91,0:09:38.13,Default,,0000,0000,0000,,So you get the square\Nroot of 10/10.
Dialogue: 0,0:09:38.13,0:09:40.63,Default,,0000,0000,0000,,These are equivalent statements,\Nbut both of them
Dialogue: 0,0:09:40.63,0:09:41.54,Default,,0000,0000,0000,,are irrational.
Dialogue: 0,0:09:41.54,0:09:43.87,Default,,0000,0000,0000,,You take an irrational number,\Ndivide it by 10, you still
Dialogue: 0,0:09:43.87,0:09:45.66,Default,,0000,0000,0000,,have an irrational number.
Dialogue: 0,0:09:45.66,0:09:46.93,Default,,0000,0000,0000,,Let's do J.
Dialogue: 0,0:09:49.52,0:09:53.82,Default,,0000,0000,0000,,We have the square\Nroot of 0.01.
Dialogue: 0,0:09:53.82,0:09:57.57,Default,,0000,0000,0000,,This is the same thing as the\Nsquare root of 1/100.
Dialogue: 0,0:09:57.57,0:10:00.68,Default,,0000,0000,0000,,Which is equal to the square\Nroot of 1 over the square root
Dialogue: 0,0:10:00.68,0:10:07.05,Default,,0000,0000,0000,,of 100, which is equal\Nto 1/10, or 0.1.
Dialogue: 0,0:10:07.05,0:10:10.03,Default,,0000,0000,0000,,Clearly once again\Nthis is rational.
Dialogue: 0,0:10:10.03,0:10:12.88,Default,,0000,0000,0000,,It's being written\Nas a fraction.
Dialogue: 0,0:10:12.88,0:10:14.18,Default,,0000,0000,0000,,This one up here was\Nalso rational.
Dialogue: 0,0:10:14.18,0:10:16.03,Default,,0000,0000,0000,,It can be written expressed\Nas a fraction.