[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.98,0:00:08.33,Default,,0000,0000,0000,,In this particular session,\Nwe're going to be looking
Dialogue: 0,0:00:08.33,0:00:11.16,Default,,0000,0000,0000,,at indices or powers.
Dialogue: 0,0:00:11.86,0:00:15.29,Default,,0000,0000,0000,,Either name is used. Both mean
Dialogue: 0,0:00:15.29,0:00:22.17,Default,,0000,0000,0000,,the same. Basically there\Na shorthand way of
Dialogue: 0,0:00:22.17,0:00:29.11,Default,,0000,0000,0000,,writing. Multiplications of the\Nsame number. So here we have 4
Dialogue: 0,0:00:29.11,0:00:35.24,Default,,0000,0000,0000,,multiplied by itself three\Ntimes. So we write that as 4
Dialogue: 0,0:00:35.24,0:00:42.48,Default,,0000,0000,0000,,to the power three, so it's\Nthree. That is the power or the
Dialogue: 0,0:00:42.48,0:00:49.09,Default,,0000,0000,0000,,index. That's the index or the\Npower. We can do this with
Dialogue: 0,0:00:49.09,0:00:55.80,Default,,0000,0000,0000,,letters, so we might have a\Ntimes a times a times a Times A
Dialogue: 0,0:00:55.80,0:01:01.06,Default,,0000,0000,0000,,and that's a multiplied by\Nitself five times. So we write
Dialogue: 0,0:01:01.06,0:01:04.42,Default,,0000,0000,0000,,that as A to the power 5.
Dialogue: 0,0:01:05.09,0:01:12.03,Default,,0000,0000,0000,,Do we have something like\N2 X squared raised to
Dialogue: 0,0:01:12.03,0:01:15.50,Default,,0000,0000,0000,,the power four, let's say?
Dialogue: 0,0:01:16.05,0:01:21.34,Default,,0000,0000,0000,,Then that would mean two X\Nsquared multiplied by two X
Dialogue: 0,0:01:21.34,0:01:26.15,Default,,0000,0000,0000,,squared multiplied by two X\Nsquared multiplied by two X
Dialogue: 0,0:01:26.15,0:01:31.92,Default,,0000,0000,0000,,squared 1234 of them all\Ntogether. So we can do the tools
Dialogue: 0,0:01:31.92,0:01:39.14,Default,,0000,0000,0000,,together. 2 * 2 * 2 * 2.\NThat gives us 16 and X squared
Dialogue: 0,0:01:39.14,0:01:42.99,Default,,0000,0000,0000,,times by X squared is X to the
Dialogue: 0,0:01:42.99,0:01:48.66,Default,,0000,0000,0000,,power 4. Times by another X\Nsquared is X to the power 6
Dialogue: 0,0:01:48.66,0:01:52.62,Default,,0000,0000,0000,,times by another. X squared is X\Nto the power 8.
Dialogue: 0,0:01:54.28,0:01:58.67,Default,,0000,0000,0000,,OK, we've got a notation. We've\Ngot a way of writing something
Dialogue: 0,0:01:58.67,0:02:02.70,Default,,0000,0000,0000,,down. Now when mathematicians\Nhave a notation and they got a
Dialogue: 0,0:02:02.70,0:02:07.46,Default,,0000,0000,0000,,way of writing something down,\Nthey want to be able to use it
Dialogue: 0,0:02:07.46,0:02:11.48,Default,,0000,0000,0000,,for other purposes. So for\Ninstance, what might A to the
Dialogue: 0,0:02:11.48,0:02:12.58,Default,,0000,0000,0000,,minus 2 mean?
Dialogue: 0,0:02:13.08,0:02:17.28,Default,,0000,0000,0000,,We know what A to the power two\Nwould mean, but what about A to
Dialogue: 0,0:02:17.28,0:02:18.96,Default,,0000,0000,0000,,the minus two? What would that
Dialogue: 0,0:02:18.96,0:02:25.40,Default,,0000,0000,0000,,mean? What would something like\NA to the power half me?
Dialogue: 0,0:02:26.53,0:02:33.00,Default,,0000,0000,0000,,What might something like A to\Nthe power 0 mean?
Dialogue: 0,0:02:33.62,0:02:39.46,Default,,0000,0000,0000,,Well, we need some rules to\Noperate with an out of looking
Dialogue: 0,0:02:39.46,0:02:43.85,Default,,0000,0000,0000,,at these rules will find what\Nthese particular notations
Dialogue: 0,0:02:43.85,0:02:49.48,Default,,0000,0000,0000,,actually mean. So let's begin\Nwith our first rule. Supposing
Dialogue: 0,0:02:49.48,0:02:56.43,Default,,0000,0000,0000,,we have a cubed and we\Nwant to multiply it by A
Dialogue: 0,0:02:56.43,0:02:58.75,Default,,0000,0000,0000,,squared, what's our result?
Dialogue: 0,0:02:59.45,0:03:05.99,Default,,0000,0000,0000,,Well, we know what I cubed is.\NThat means a times a times a
Dialogue: 0,0:03:05.99,0:03:12.53,Default,,0000,0000,0000,,three times times by A squared.\NSo that's a times by a on the
Dialogue: 0,0:03:12.53,0:03:18.60,Default,,0000,0000,0000,,end there, and altogether we've\Ngot five of them A to the power
Dialogue: 0,0:03:18.60,0:03:24.87,Default,,0000,0000,0000,,5. And that suggests our very\Nfirst rule that if we're
Dialogue: 0,0:03:24.87,0:03:30.02,Default,,0000,0000,0000,,multiplying together expressions\Nsuch as these, then we add the
Dialogue: 0,0:03:30.02,0:03:37.23,Default,,0000,0000,0000,,indices and so if we have A\Nto the M times by A to
Dialogue: 0,0:03:37.23,0:03:44.44,Default,,0000,0000,0000,,the N and the result is A\Nto the N plus N, and that's
Dialogue: 0,0:03:44.44,0:03:45.98,Default,,0000,0000,0000,,our first rule.
Dialogue: 0,0:03:46.69,0:03:51.54,Default,,0000,0000,0000,,Let's have a look at our second\Nrule. Already done something
Dialogue: 0,0:03:51.54,0:03:56.83,Default,,0000,0000,0000,,like this previously. Supposing\Nwe had A to the power four, and
Dialogue: 0,0:03:56.83,0:04:03.01,Default,,0000,0000,0000,,we want to raise it all to the\Npower three, and we know what
Dialogue: 0,0:04:03.01,0:04:09.18,Default,,0000,0000,0000,,that means. It means A to the\Npower four times by A to the
Dialogue: 0,0:04:09.18,0:04:12.27,Default,,0000,0000,0000,,power four times by A to the
Dialogue: 0,0:04:12.27,0:04:19.14,Default,,0000,0000,0000,,power 4. Now the first rule\Ntells us that we should add the
Dialogue: 0,0:04:19.14,0:04:26.06,Default,,0000,0000,0000,,indices together, so that's A to\Nthe power twelve. 4 + 4 + 4.
Dialogue: 0,0:04:26.99,0:04:32.80,Default,,0000,0000,0000,,But twelve is 3 times by 4, so\Nthat suggests to us that we
Dialogue: 0,0:04:32.80,0:04:36.95,Default,,0000,0000,0000,,should, perhaps, if we've got A\Nto the power M.
Dialogue: 0,0:04:37.85,0:04:44.67,Default,,0000,0000,0000,,Raised to the power N,\Nthen the result we get
Dialogue: 0,0:04:44.67,0:04:49.44,Default,,0000,0000,0000,,by multiplying those two\Ntogether and that.
Dialogue: 0,0:04:50.12,0:04:52.81,Default,,0000,0000,0000,,Is our second rule.
Dialogue: 0,0:04:53.71,0:04:57.66,Default,,0000,0000,0000,,Let's now have a look.
Dialogue: 0,0:04:58.34,0:05:00.89,Default,,0000,0000,0000,,And our third rule.
Dialogue: 0,0:05:01.63,0:05:05.46,Default,,0000,0000,0000,,For this, let's take A to the
Dialogue: 0,0:05:05.46,0:05:12.70,Default,,0000,0000,0000,,7th. And let's divide it by\NA to the power three or a
Dialogue: 0,0:05:12.70,0:05:18.30,Default,,0000,0000,0000,,cubed well, A to the 7th means\Na multiplied by itself.
Dialogue: 0,0:05:19.12,0:05:26.18,Default,,0000,0000,0000,,Seven times.\NDivided by so let's divide it
Dialogue: 0,0:05:26.18,0:05:31.98,Default,,0000,0000,0000,,by. A multiplied by itself 3\Ntimes and now we can begin to
Dialogue: 0,0:05:31.98,0:05:36.72,Default,,0000,0000,0000,,cancel some common factors. So\Nthere's a common factor of A and
Dialogue: 0,0:05:36.72,0:05:40.67,Default,,0000,0000,0000,,again, there's another common\Nfactor of A and again, there's
Dialogue: 0,0:05:40.67,0:05:45.41,Default,,0000,0000,0000,,another common factor of a. So\Non the bottom here we've really
Dialogue: 0,0:05:45.41,0:05:53.02,Default,,0000,0000,0000,,got. 1 one and one suite. 1 *\N1 is one and on the top a Times
Dialogue: 0,0:05:53.02,0:05:57.88,Default,,0000,0000,0000,,by a times by a Times by AA to\Nthe power 4.
Dialogue: 0,0:05:58.45,0:06:04.79,Default,,0000,0000,0000,,But Seven takeaway three is A to\Nthe power four, and so that
Dialogue: 0,0:06:04.79,0:06:12.11,Default,,0000,0000,0000,,gives us our third rule that if\Nwe have A to the power M divided
Dialogue: 0,0:06:12.11,0:06:19.43,Default,,0000,0000,0000,,by A to the power N, we get\Nthe result A to the power M
Dialogue: 0,0:06:19.43,0:06:25.85,Default,,0000,0000,0000,,minus N. And so there's\Nour third rule.
Dialogue: 0,0:06:26.39,0:06:31.84,Default,,0000,0000,0000,,OK, we got 3 rules. Let's see\Nwhat we can do with them.
Dialogue: 0,0:06:34.34,0:06:42.01,Default,,0000,0000,0000,,Let's have a look\Nat a cubed divided
Dialogue: 0,0:06:42.01,0:06:44.89,Default,,0000,0000,0000,,by a cubed.
Dialogue: 0,0:06:45.80,0:06:51.27,Default,,0000,0000,0000,,When we know the answer to that,\Na cubed divided by a cube we're
Dialogue: 0,0:06:51.27,0:06:55.97,Default,,0000,0000,0000,,dividing something by itself. So\Nthe answer is got to be 1.
Dialogue: 0,0:06:57.13,0:07:03.40,Default,,0000,0000,0000,,Fine, let's do it using our laws\Nof indices, our rules, and we
Dialogue: 0,0:07:03.40,0:07:10.63,Default,,0000,0000,0000,,can use Rule #3 for this that if\Nwe want to do this, we subtract
Dialogue: 0,0:07:10.63,0:07:17.86,Default,,0000,0000,0000,,the indices. So that's A to the\Npower 3 - 3, which is A to
Dialogue: 0,0:07:17.86,0:07:19.30,Default,,0000,0000,0000,,the power 0.
Dialogue: 0,0:07:20.24,0:07:25.78,Default,,0000,0000,0000,,So what have we done? We've done\Nthe same calculation in two
Dialogue: 0,0:07:25.78,0:07:30.40,Default,,0000,0000,0000,,different ways. We've done it\Ncorrectly in two different ways,
Dialogue: 0,0:07:30.40,0:07:36.41,Default,,0000,0000,0000,,so the answers that we get, even\Nif they look different, must be
Dialogue: 0,0:07:36.41,0:07:43.34,Default,,0000,0000,0000,,the same. And So what we have is\NA to the power 0 equals 1.
Dialogue: 0,0:07:43.37,0:07:48.25,Default,,0000,0000,0000,,Our 4th result. If you like what\Ndoes this mean?
Dialogue: 0,0:07:48.92,0:07:53.25,Default,,0000,0000,0000,,Any fact it means that any\Nnumber raised to the power zero
Dialogue: 0,0:07:53.25,0:07:59.03,Default,,0000,0000,0000,,is one. So if we take two as we\Nraise it to the power zero, the
Dialogue: 0,0:07:59.03,0:08:00.11,Default,,0000,0000,0000,,answer is 1.
Dialogue: 0,0:08:00.64,0:08:07.26,Default,,0000,0000,0000,,If we take a million and\Nraise it to the power zero,
Dialogue: 0,0:08:07.26,0:08:09.47,Default,,0000,0000,0000,,the answer is 1.
Dialogue: 0,0:08:10.94,0:08:17.48,Default,,0000,0000,0000,,If we take something like a half\Nand raise it to the power zero,
Dialogue: 0,0:08:17.48,0:08:24.02,Default,,0000,0000,0000,,the answer is again one we take\Nminus six and raise it to the
Dialogue: 0,0:08:24.02,0:08:26.35,Default,,0000,0000,0000,,power zero. The answers one.
Dialogue: 0,0:08:27.68,0:08:30.33,Default,,0000,0000,0000,,If we take zero and raise it to
Dialogue: 0,0:08:30.33,0:08:35.77,Default,,0000,0000,0000,,the power 0. Well, it's a bit\Ncomplicated, so we'll leave that
Dialogue: 0,0:08:35.77,0:08:41.45,Default,,0000,0000,0000,,one on side for the moment. Just\Nbear in mind any number apart
Dialogue: 0,0:08:41.45,0:08:46.70,Default,,0000,0000,0000,,from zero when raised to the\Npower zero is equal to 1.
Dialogue: 0,0:08:47.39,0:08:54.44,Default,,0000,0000,0000,,Let's have a look\Nnow at doing a
Dialogue: 0,0:08:54.44,0:08:55.32,Default,,0000,0000,0000,,division.
Dialogue: 0,0:08:56.50,0:09:01.37,Default,,0000,0000,0000,,Again. Let's take the example\Nthat we use when we looked at
Dialogue: 0,0:09:01.37,0:09:05.01,Default,,0000,0000,0000,,law three, except let's turn it\Nround, let's do the division the
Dialogue: 0,0:09:05.01,0:09:10.05,Default,,0000,0000,0000,,other way about. A cubed divided\Nby A to the 7th.
Dialogue: 0,0:09:10.72,0:09:15.93,Default,,0000,0000,0000,,Well, you can set that out as we\Ndid before, except it will be
Dialogue: 0,0:09:15.93,0:09:22.25,Default,,0000,0000,0000,,the other way up. So we have a\Ncubed is a Times by a times by a
Dialogue: 0,0:09:22.25,0:09:26.07,Default,,0000,0000,0000,,divided by. A multiplied
Dialogue: 0,0:09:26.07,0:09:29.68,Default,,0000,0000,0000,,by itself. 7.
Dialogue: 0,0:09:30.39,0:09:36.67,Default,,0000,0000,0000,,Times. Again, we can do\Nthe canceling, canceling out the
Dialogue: 0,0:09:36.67,0:09:41.66,Default,,0000,0000,0000,,common factors, dividing top and\Nbottom by the common factors.
Dialogue: 0,0:09:42.53,0:09:49.66,Default,,0000,0000,0000,,So what do we have? One on\Nthe top 1234 on the bottom A
Dialogue: 0,0:09:49.66,0:09:51.69,Default,,0000,0000,0000,,to the Power 4?
Dialogue: 0,0:09:52.31,0:09:59.94,Default,,0000,0000,0000,,We know we've done that right?\NLet's use our third law, our
Dialogue: 0,0:09:59.94,0:10:05.67,Default,,0000,0000,0000,,third rule, and do it by\Nsubtracting the indices.
Dialogue: 0,0:10:06.13,0:10:13.50,Default,,0000,0000,0000,,Well, three takeaway 7 is minus\Nfour, so we've got A to the
Dialogue: 0,0:10:13.50,0:10:19.17,Default,,0000,0000,0000,,power minus four. So same\Nargument applies. We've done the
Dialogue: 0,0:10:19.17,0:10:24.49,Default,,0000,0000,0000,,calculation. Same calculation in\Ntwo different ways.
Dialogue: 0,0:10:25.16,0:10:30.25,Default,,0000,0000,0000,,We've done it correctly. We've\Narrived at two different answers
Dialogue: 0,0:10:30.25,0:10:36.36,Default,,0000,0000,0000,,there for these two answers.\NHave got to be the same. So
Dialogue: 0,0:10:36.36,0:10:42.98,Default,,0000,0000,0000,,one over 8 to the power four\Nis a till they minus 4.
Dialogue: 0,0:10:43.94,0:10:50.22,Default,,0000,0000,0000,,So a minus sign in the index\Nwith the power means one over
Dialogue: 0,0:10:50.22,0:10:56.50,Default,,0000,0000,0000,,one over A to the power four.\NLet's just develop that one a
Dialogue: 0,0:10:56.50,0:11:02.29,Default,,0000,0000,0000,,little bit. Let's just look at\None or two examples. So for
Dialogue: 0,0:11:02.29,0:11:07.61,Default,,0000,0000,0000,,instance, 2 to the power minus\Ntwo would be one over.
Dialogue: 0,0:11:08.20,0:11:14.98,Default,,0000,0000,0000,,Two Square, which of course\Ngives us a quarter.
Dialogue: 0,0:11:15.88,0:11:22.78,Default,,0000,0000,0000,,5 to the power minus one\Nis one over 5 to the
Dialogue: 0,0:11:22.78,0:11:26.80,Default,,0000,0000,0000,,one which is just one\Nover 5.
Dialogue: 0,0:11:27.98,0:11:30.00,Default,,0000,0000,0000,,One
Dialogue: 0,0:11:30.00,0:11:38.50,Default,,0000,0000,0000,,over. Hey.\NIs A to the minus one turning
Dialogue: 0,0:11:38.50,0:11:40.78,Default,,0000,0000,0000,,it round working backwards?
Dialogue: 0,0:11:41.52,0:11:48.52,Default,,0000,0000,0000,,One over 7 squared would be\None over 49, but what about
Dialogue: 0,0:11:48.52,0:11:55.51,Default,,0000,0000,0000,,one over 7 to the minus\N2 - 2 remember means one
Dialogue: 0,0:11:55.51,0:12:02.51,Default,,0000,0000,0000,,over 7 squared's. This is one\Nover one over 7 squared, and
Dialogue: 0,0:12:02.51,0:12:08.92,Default,,0000,0000,0000,,here we're dividing by a\Nfraction and to divide by a
Dialogue: 0,0:12:08.92,0:12:11.84,Default,,0000,0000,0000,,fraction, we know that we.
Dialogue: 0,0:12:11.86,0:12:18.82,Default,,0000,0000,0000,,Invert and multiply and so 7\Ntimes by 7 is 49 times
Dialogue: 0,0:12:18.82,0:12:25.20,Default,,0000,0000,0000,,by the one leaves us with\N49 or just 7 squared.
Dialogue: 0,0:12:26.46,0:12:31.19,Default,,0000,0000,0000,,So some examples there. This is\Nprobably the one that you need
Dialogue: 0,0:12:31.19,0:12:35.92,Default,,0000,0000,0000,,to remember and need to work\Nwith most. It's the basic case
Dialogue: 0,0:12:35.92,0:12:41.04,Default,,0000,0000,0000,,and if you can remember that one\Nthen they nearly all follow from
Dialogue: 0,0:12:41.04,0:12:44.58,Default,,0000,0000,0000,,that. So that's that one. Let's\Nnow go on.
Dialogue: 0,0:12:45.14,0:12:47.54,Default,,0000,0000,0000,,And have a look.
Dialogue: 0,0:12:47.62,0:12:50.02,Default,,0000,0000,0000,,A tower 6 result.
Dialogue: 0,0:12:50.55,0:12:57.02,Default,,0000,0000,0000,,What do we mean\Nby A to the
Dialogue: 0,0:12:57.02,0:13:02.90,Default,,0000,0000,0000,,power 1/2? What's that mean? So\Nfar we've been working with
Dialogue: 0,0:13:02.90,0:13:04.53,Default,,0000,0000,0000,,integers an with negative
Dialogue: 0,0:13:04.53,0:13:09.48,Default,,0000,0000,0000,,numbers. What about A to the\Npower 1/2 well?
Dialogue: 0,0:13:10.23,0:13:16.63,Default,,0000,0000,0000,,Supposing we had A to the P and\Nwe multiplied it by A to the P,
Dialogue: 0,0:13:16.63,0:13:19.43,Default,,0000,0000,0000,,the answer we got was just a.
Dialogue: 0,0:13:20.03,0:13:23.73,Default,,0000,0000,0000,,Just a That's A to the
Dialogue: 0,0:13:23.73,0:13:27.25,Default,,0000,0000,0000,,power one. And a Times
Dialogue: 0,0:13:27.25,0:13:33.64,Default,,0000,0000,0000,,by a. Each with a P on A to the\NP times by A to the P using our
Dialogue: 0,0:13:33.64,0:13:35.64,Default,,0000,0000,0000,,rule would be A to the 2P.
Dialogue: 0,0:13:36.83,0:13:43.65,Default,,0000,0000,0000,,So 2P must be the same as one.\NIn other words, P is 1/2.
Dialogue: 0,0:13:44.66,0:13:49.46,Default,,0000,0000,0000,,What do we have some sort of\Ninterpretation for? This two
Dialogue: 0,0:13:49.46,0:13:53.82,Default,,0000,0000,0000,,numbers, identical that multiply\Ntogether to give a? Well, that's
Dialogue: 0,0:13:53.82,0:13:55.100,Default,,0000,0000,0000,,the square root, isn't it?
Dialogue: 0,0:13:56.66,0:14:02.42,Default,,0000,0000,0000,,It's a square root. It's like 7\Ntimes by 7 equals 49.
Dialogue: 0,0:14:03.39,0:14:05.77,Default,,0000,0000,0000,,So if we take that one on.
Dialogue: 0,0:14:06.90,0:14:14.46,Default,,0000,0000,0000,,7 times by 7 is 49. What\Nwe've got then is 49 to the
Dialogue: 0,0:14:14.46,0:14:22.02,Default,,0000,0000,0000,,half is equal to 7, so ETA\Nthe half is equal to the square
Dialogue: 0,0:14:22.02,0:14:23.64,Default,,0000,0000,0000,,root of A.
Dialogue: 0,0:14:24.19,0:14:31.61,Default,,0000,0000,0000,,A to the third would be equal\Nto the cube root of A.
Dialogue: 0,0:14:32.21,0:14:38.76,Default,,0000,0000,0000,,So if we were asked what\Nis 16 to the quarter?
Dialogue: 0,0:14:38.76,0:14:42.99,Default,,0000,0000,0000,,What we're asking is what\Nnumber, when multiplied by
Dialogue: 0,0:14:42.99,0:14:48.63,Default,,0000,0000,0000,,itself four times, gives us 16.\NA fairly obvious choice for that
Dialogue: 0,0:14:48.63,0:14:51.52,Default,,0000,0000,0000,,is 2. What
Dialogue: 0,0:14:51.52,0:14:57.61,Default,,0000,0000,0000,,about 81? To\Nthe half? Well, that's the
Dialogue: 0,0:14:57.61,0:15:03.16,Default,,0000,0000,0000,,square root of 81, and the\Nsquare root of 81 is 9.
Dialogue: 0,0:15:03.96,0:15:10.87,Default,,0000,0000,0000,,What about 243\Nand will make
Dialogue: 0,0:15:10.87,0:15:14.33,Default,,0000,0000,0000,,that to the
Dialogue: 0,0:15:14.33,0:15:19.99,Default,,0000,0000,0000,,5th? What number, when\Nmultiplied by itself five times,
Dialogue: 0,0:15:19.99,0:15:27.42,Default,,0000,0000,0000,,will give us 240 three? Well, as\Nwe look at this, we can see it
Dialogue: 0,0:15:27.42,0:15:33.85,Default,,0000,0000,0000,,divides by three and three's\Ninto that give us 81, so we know
Dialogue: 0,0:15:33.85,0:15:36.82,Default,,0000,0000,0000,,this is 3 times by 81.
Dialogue: 0,0:15:37.41,0:15:41.18,Default,,0000,0000,0000,,81 we know.
Dialogue: 0,0:15:41.79,0:15:45.55,Default,,0000,0000,0000,,Is 9 times by 9.
Dialogue: 0,0:15:45.55,0:15:52.20,Default,,0000,0000,0000,,And each of\Nthose nines is
Dialogue: 0,0:15:52.20,0:15:55.52,Default,,0000,0000,0000,,3 times by
Dialogue: 0,0:15:55.52,0:16:02.40,Default,,0000,0000,0000,,three. Which means the number\Nthat we want is just three.
Dialogue: 0,0:16:03.26,0:16:09.35,Default,,0000,0000,0000,,Noticing doing this, how\Nimportant it is to be able to
Dialogue: 0,0:16:09.35,0:16:16.00,Default,,0000,0000,0000,,recognize what numbers are made\Nup of to be able to recognize
Dialogue: 0,0:16:16.00,0:16:23.76,Default,,0000,0000,0000,,that 16 is 2 to the power,\Nfour that it's also 4 to the
Dialogue: 0,0:16:23.76,0:16:31.51,Default,,0000,0000,0000,,power to the nine is 3 squared.\NThat 81 is 9 squared and also
Dialogue: 0,0:16:31.51,0:16:34.28,Default,,0000,0000,0000,,3 to the power 4.
Dialogue: 0,0:16:34.32,0:16:40.39,Default,,0000,0000,0000,,You'll find calculations much,\Nmuch easier if you can recognize
Dialogue: 0,0:16:40.39,0:16:47.07,Default,,0000,0000,0000,,in numbers their composition as\Npowers of simple numbers such as
Dialogue: 0,0:16:47.07,0:16:50.71,Default,,0000,0000,0000,,two and three, four, and five.
Dialogue: 0,0:16:51.22,0:16:53.77,Default,,0000,0000,0000,,Once you've got those firmly\Nfixed in your mind.
Dialogue: 0,0:16:54.82,0:16:58.32,Default,,0000,0000,0000,,This sort of calculation becomes
Dialogue: 0,0:16:58.32,0:17:01.15,Default,,0000,0000,0000,,relatively straightforward. 1
Dialogue: 0,0:17:01.15,0:17:07.51,Default,,0000,0000,0000,,final result. If\Nwe now know what A to the half
Dialogue: 0,0:17:07.51,0:17:12.23,Default,,0000,0000,0000,,isn't A to the third and eight\Nto the quarter, what do we mean
Dialogue: 0,0:17:12.23,0:17:14.59,Default,,0000,0000,0000,,if we take A to the 3/4?
Dialogue: 0,0:17:15.80,0:17:21.30,Default,,0000,0000,0000,,Well, the quarters alright.\NLet's split that up. That means
Dialogue: 0,0:17:21.30,0:17:24.05,Default,,0000,0000,0000,,A to the power 1/4.
Dialogue: 0,0:17:24.97,0:17:29.25,Default,,0000,0000,0000,,Cubed and what we're doing is\Nwe're using this result that A
Dialogue: 0,0:17:29.25,0:17:34.97,Default,,0000,0000,0000,,to the M raised to the power N\Nis A to the MN. In other words,
Dialogue: 0,0:17:34.97,0:17:38.89,Default,,0000,0000,0000,,we're using our 2nd result to be\Nable to do that.
Dialogue: 0,0:17:40.19,0:17:46.99,Default,,0000,0000,0000,,So let's have a look at an\Nexample using this. Supposing we
Dialogue: 0,0:17:46.99,0:17:53.23,Default,,0000,0000,0000,,take 60, we say we want 16\Nto the power 3/4.
Dialogue: 0,0:17:53.24,0:17:56.73,Default,,0000,0000,0000,,Then that 16 to the power
Dialogue: 0,0:17:56.73,0:18:03.98,Default,,0000,0000,0000,,quarter. To be cubed, so we\Nlook at this first 16 to the
Dialogue: 0,0:18:03.98,0:18:09.58,Default,,0000,0000,0000,,Power 1/4 is 2. That's the\Nnumber when multiplied by itself
Dialogue: 0,0:18:09.58,0:18:15.18,Default,,0000,0000,0000,,four times will give us 16\Nraised to the power 3.
Dialogue: 0,0:18:15.19,0:18:19.60,Default,,0000,0000,0000,,And that of course is 8 because\Nthat means 2 * 2 * 2.
Dialogue: 0,0:18:20.22,0:18:25.10,Default,,0000,0000,0000,,But we can think of this another\Nway, because Eminem can be
Dialogue: 0,0:18:25.10,0:18:29.99,Default,,0000,0000,0000,,interchanged. Currently we could\Nwrite this as A to the power N
Dialogue: 0,0:18:29.99,0:18:32.02,Default,,0000,0000,0000,,raised to the power M.
Dialogue: 0,0:18:32.04,0:18:39.36,Default,,0000,0000,0000,,That would be just the same\Nresult, so we can think of this
Dialogue: 0,0:18:39.36,0:18:44.99,Default,,0000,0000,0000,,in a slightly different way.\NLet's take a different example.
Dialogue: 0,0:18:44.99,0:18:46.68,Default,,0000,0000,0000,,Let's take 8.
Dialogue: 0,0:18:46.69,0:18:49.79,Default,,0000,0000,0000,,To the power 2/3.
Dialogue: 0,0:18:50.29,0:18:56.60,Default,,0000,0000,0000,,Now, the way that we had thought\Nabout that was to do 8 to the
Dialogue: 0,0:18:56.60,0:18:58.91,Default,,0000,0000,0000,,power third. Then square it.
Dialogue: 0,0:18:59.49,0:19:05.66,Default,,0000,0000,0000,,And we know that 8 to the\Npower 1:30 is 2 and 2
Dialogue: 0,0:19:05.66,0:19:08.04,Default,,0000,0000,0000,,squared is there for four.
Dialogue: 0,0:19:09.13,0:19:15.47,Default,,0000,0000,0000,,We don't have to think of it\Nlike that. We can think of it
Dialogue: 0,0:19:15.47,0:19:20.46,Default,,0000,0000,0000,,as 8 squared raised to the\NPower, 1/3 eight squared we
Dialogue: 0,0:19:20.46,0:19:27.25,Default,,0000,0000,0000,,know is 64 and now we have to\Ntake it to the power 1/3. We
Dialogue: 0,0:19:27.25,0:19:32.23,Default,,0000,0000,0000,,need the cube root. We need a\Nnumber that when multiplied
Dialogue: 0,0:19:32.23,0:19:37.67,Default,,0000,0000,0000,,by itself three times gives\Nus 64 and that number is 4.
Dialogue: 0,0:19:38.75,0:19:41.41,Default,,0000,0000,0000,,These two are equivalent.
Dialogue: 0,0:19:41.93,0:19:47.69,Default,,0000,0000,0000,,So the different interpretations\Nthat we've got are the same.
Dialogue: 0,0:19:47.69,0:19:52.06,Default,,0000,0000,0000,,Writing it down algebraically,\Nwhat we're saying is that we
Dialogue: 0,0:19:52.06,0:19:58.94,Default,,0000,0000,0000,,have. A letter of variable to\Nthe power P over Q. Then we can
Dialogue: 0,0:19:58.94,0:20:05.35,Default,,0000,0000,0000,,write that in one of two ways.\NWe can write it as either A to
Dialogue: 0,0:20:05.35,0:20:06.63,Default,,0000,0000,0000,,the power P.
Dialogue: 0,0:20:07.14,0:20:14.31,Default,,0000,0000,0000,,Find the cube root of it, which\Nmight be written as A to the
Dialogue: 0,0:20:14.31,0:20:21.48,Default,,0000,0000,0000,,power P. Find the cube root, or\Nit might be written as a. Let's
Dialogue: 0,0:20:21.48,0:20:27.62,Default,,0000,0000,0000,,take the cube root and raise it\Nall to the power P.
Dialogue: 0,0:20:28.16,0:20:34.66,Default,,0000,0000,0000,,So we might have a find the cube\Nroot of it and raise it all to
Dialogue: 0,0:20:34.66,0:20:35.87,Default,,0000,0000,0000,,the power P.
Dialogue: 0,0:20:36.42,0:20:39.46,Default,,0000,0000,0000,,Either result is exactly the
Dialogue: 0,0:20:39.46,0:20:46.29,Default,,0000,0000,0000,,same. Now, to conclude, let's\Njust have a look at some very
Dialogue: 0,0:20:46.29,0:20:48.23,Default,,0000,0000,0000,,basic simple calculations using
Dialogue: 0,0:20:48.23,0:20:54.54,Default,,0000,0000,0000,,indices. 2X to the minus 1/4 and\Nour objective here is to write
Dialogue: 0,0:20:54.54,0:20:59.73,Default,,0000,0000,0000,,it with a positive index. Well,\Nthe two doesn't have an index
Dialogue: 0,0:20:59.73,0:21:04.93,Default,,0000,0000,0000,,attached to it at all, so\Nnothing happens to the two just
Dialogue: 0,0:21:04.93,0:21:10.99,Default,,0000,0000,0000,,stays it is X to the minus 1/4.\NThe minus sign means one over
Dialogue: 0,0:21:10.99,0:21:16.62,Default,,0000,0000,0000,,right it in the denominator, so\Nwe write it down there. So we
Dialogue: 0,0:21:16.62,0:21:19.65,Default,,0000,0000,0000,,have two over X to the power
Dialogue: 0,0:21:19.65,0:21:25.17,Default,,0000,0000,0000,,quarter. 4X to the minus 2A\Ncubed. Well, nothing wrong with
Dialogue: 0,0:21:25.17,0:21:31.19,Default,,0000,0000,0000,,the four and nothing wrong with\Nthe A cubed at perfectly normal
Dialogue: 0,0:21:31.19,0:21:34.21,Default,,0000,0000,0000,,so they stay as they are.
Dialogue: 0,0:21:34.75,0:21:36.33,Default,,0000,0000,0000,,X to the minus 2.
Dialogue: 0,0:21:37.23,0:21:44.71,Default,,0000,0000,0000,,A minus in the index and so\Nthat means one over, so it's 4A
Dialogue: 0,0:21:44.71,0:21:46.84,Default,,0000,0000,0000,,cubed over X squared.
Dialogue: 0,0:21:48.85,0:21:54.55,Default,,0000,0000,0000,,One over 4A to the\Npower minus 2.
Dialogue: 0,0:21:55.28,0:21:58.84,Default,,0000,0000,0000,,Well, we met this one before.
Dialogue: 0,0:21:59.34,0:22:06.13,Default,,0000,0000,0000,,If you remember, we actually\Nlooked at one over 7 to the
Dialogue: 0,0:22:06.13,0:22:13.44,Default,,0000,0000,0000,,minus 2. And that was\None over one over 7 squared
Dialogue: 0,0:22:13.44,0:22:20.46,Default,,0000,0000,0000,,and because we were dividing by\Na fraction, we said we had
Dialogue: 0,0:22:20.46,0:22:27.48,Default,,0000,0000,0000,,to invert and multiply. And so\Nwe ended up with 7 squared.
Dialogue: 0,0:22:28.44,0:22:33.06,Default,,0000,0000,0000,,This is no different. As a\Nletter there instead of a
Dialogue: 0,0:22:33.06,0:22:37.26,Default,,0000,0000,0000,,number, the result is going to\Nbe exactly the same.
Dialogue: 0,0:22:37.83,0:22:44.94,Default,,0000,0000,0000,,So the four stays where it is\None over 4 and this becomes a
Dialogue: 0,0:22:44.94,0:22:50.53,Default,,0000,0000,0000,,squared and to write it in a\Nmore simple, tidier fashion,
Dialogue: 0,0:22:50.53,0:22:53.07,Default,,0000,0000,0000,,it's A squared over 4.
Dialogue: 0,0:22:54.13,0:23:01.80,Default,,0000,0000,0000,,A to the minus 1/3\Ntimes by two A to
Dialogue: 0,0:23:01.80,0:23:04.87,Default,,0000,0000,0000,,the minus 1/2 equals.
Dialogue: 0,0:23:05.48,0:23:13.14,Default,,0000,0000,0000,,Two, we can just write down A to\Nthe minus 1/3 times by 8 in the
Dialogue: 0,0:23:13.14,0:23:18.89,Default,,0000,0000,0000,,minus 1/2 hour. First job is to\Nadd those indices together, so
Dialogue: 0,0:23:18.89,0:23:21.29,Default,,0000,0000,0000,,minus 1/3 plus minus 1/2.
Dialogue: 0,0:23:21.93,0:23:29.60,Default,,0000,0000,0000,,He's going to give us 2A to\Nthe power now. A third and a
Dialogue: 0,0:23:29.60,0:23:36.73,Default,,0000,0000,0000,,half is 5 six. So with the\Nminus signs, that is minus five
Dialogue: 0,0:23:36.73,0:23:43.30,Default,,0000,0000,0000,,sixths, and so that's two over A\Nto the power five sixths.
Dialogue: 0,0:23:44.27,0:23:50.77,Default,,0000,0000,0000,,2A to the\Nminus 2 divided
Dialogue: 0,0:23:50.77,0:23:57.28,Default,,0000,0000,0000,,by A to\Nthe minus three
Dialogue: 0,0:23:57.28,0:24:04.79,Default,,0000,0000,0000,,over 2. Well, that's\N2A to the minus 2 divided
Dialogue: 0,0:24:04.79,0:24:08.35,Default,,0000,0000,0000,,by A to the minus three
Dialogue: 0,0:24:08.35,0:24:14.58,Default,,0000,0000,0000,,over 2. Let's remember what our\Nrules said that if we're
Dialogue: 0,0:24:14.58,0:24:18.96,Default,,0000,0000,0000,,dividing things like this, we\Nactually subtract the indices.
Dialogue: 0,0:24:18.96,0:24:25.76,Default,,0000,0000,0000,,So this is 2A to the minus 2\Nminus minus three over two. Of
Dialogue: 0,0:24:25.76,0:24:31.11,Default,,0000,0000,0000,,course, that means effectively\Nwe've got to add on to three
Dialogue: 0,0:24:31.11,0:24:38.88,Default,,0000,0000,0000,,over 2, so we get 2A to the\Nminus 2 + 3 over 2 inches minus
Dialogue: 0,0:24:38.88,0:24:40.83,Default,,0000,0000,0000,,1/2, so that's 2A.
Dialogue: 0,0:24:40.84,0:24:44.24,Default,,0000,0000,0000,,Over A to the half.
Dialogue: 0,0:24:44.79,0:24:52.33,Default,,0000,0000,0000,,Now then we take\Nsomething that looks a
Dialogue: 0,0:24:52.33,0:24:55.15,Default,,0000,0000,0000,,little bit complicated.
Dialogue: 0,0:24:55.82,0:25:03.37,Default,,0000,0000,0000,,So what do we got\Nhere? We've got the cube
Dialogue: 0,0:25:03.37,0:25:10.92,Default,,0000,0000,0000,,root of A squared times\Nby the square root of
Dialogue: 0,0:25:10.92,0:25:17.56,Default,,0000,0000,0000,,a cube. First of all, let's\Nwrite these as indices. This is
Dialogue: 0,0:25:17.56,0:25:23.40,Default,,0000,0000,0000,,the cube root of A squared. So\Nthat means a squared and take
Dialogue: 0,0:25:23.40,0:25:27.89,Default,,0000,0000,0000,,the cube root. You know it's A\Nto the 2/3.
Dialogue: 0,0:25:28.52,0:25:35.54,Default,,0000,0000,0000,,Times by a now this is\Na cubed. Take the square root
Dialogue: 0,0:25:35.54,0:25:39.05,Default,,0000,0000,0000,,so that's a cubed. Take the
Dialogue: 0,0:25:39.05,0:25:46.40,Default,,0000,0000,0000,,square root. And we can see\Nnow what we need to do is add
Dialogue: 0,0:25:46.40,0:25:52.35,Default,,0000,0000,0000,,together these two indices. So\Nthat's 2/3 + 3 over 2 and adding
Dialogue: 0,0:25:52.35,0:25:57.85,Default,,0000,0000,0000,,fractions together. That will\Ngive us A to the power 13 over
Dialogue: 0,0:25:57.85,0:26:03.80,Default,,0000,0000,0000,,6 awkward number will just leave\Nit like that for now. Now let's
Dialogue: 0,0:26:03.80,0:26:09.30,Default,,0000,0000,0000,,have a look at some calculations\Nusing numbers. This time 16 to
Dialogue: 0,0:26:09.30,0:26:13.88,Default,,0000,0000,0000,,the Power 3/4. Now again, we've\Nalready done this one.
Dialogue: 0,0:26:14.81,0:26:18.56,Default,,0000,0000,0000,,That 16 to the power 1/4.
Dialogue: 0,0:26:19.12,0:26:23.72,Default,,0000,0000,0000,,Cubed what number, when\Nmultiplied by itself four times,
Dialogue: 0,0:26:23.72,0:26:30.36,Default,,0000,0000,0000,,gives us 16? That's two raised\Nto the power. Three gives us 8
Dialogue: 0,0:26:30.36,0:26:31.90,Default,,0000,0000,0000,,for our answer.
Dialogue: 0,0:26:32.68,0:26:39.58,Default,,0000,0000,0000,,4 to the power minus five over\N2. Let's deal with the minus
Dialogue: 0,0:26:39.58,0:26:45.42,Default,,0000,0000,0000,,sign first. Minus sign means one\Nover. Put it until the
Dialogue: 0,0:26:45.42,0:26:48.61,Default,,0000,0000,0000,,denominator, 4 to the power 5
Dialogue: 0,0:26:48.61,0:26:55.08,Default,,0000,0000,0000,,over 2. One over 4 to\Nthe half to the power five
Dialogue: 0,0:26:55.08,0:27:00.92,Default,,0000,0000,0000,,notice each time I'm doing the\Nroot bit first, the 4th root
Dialogue: 0,0:27:00.92,0:27:05.79,Default,,0000,0000,0000,,here, the square root here.\NThat's because by doing the
Dialogue: 0,0:27:05.79,0:27:11.15,Default,,0000,0000,0000,,route, I get the number smaller.\NI can handle the arithmetic
Dialogue: 0,0:27:11.15,0:27:18.46,Default,,0000,0000,0000,,easier, so this is now one over\Nthe square root of 4 is 2 raised
Dialogue: 0,0:27:18.46,0:27:20.40,Default,,0000,0000,0000,,to the power 5.
Dialogue: 0,0:27:20.44,0:27:24.39,Default,,0000,0000,0000,,And that's one over 32 again.\NNotice I wrote it straight down
Dialogue: 0,0:27:24.39,0:27:28.34,Default,,0000,0000,0000,,without seeming to work it out.\NThat's because I know what it
Dialogue: 0,0:27:28.34,0:27:32.28,Default,,0000,0000,0000,,is. And again, it's another one\Nof these relationships. 2 to the
Dialogue: 0,0:27:32.28,0:27:36.89,Default,,0000,0000,0000,,power, three is 8 two to the\Npower, five is 32 to the power,
Dialogue: 0,0:27:36.89,0:27:40.84,Default,,0000,0000,0000,,six is 64 that really, if you\Ncan learn and get become
Dialogue: 0,0:27:40.84,0:27:44.13,Default,,0000,0000,0000,,familiar with you, find this\Nkind of calculation much easier.
Dialogue: 0,0:27:45.61,0:27:53.42,Default,,0000,0000,0000,,Here's another one. 125 seems\Na very big number to
Dialogue: 0,0:27:53.42,0:28:00.55,Default,,0000,0000,0000,,the 2/3. Equals 125\Nto the power of
Dialogue: 0,0:28:00.55,0:28:03.05,Default,,0000,0000,0000,,3rd all squared.
Dialogue: 0,0:28:03.75,0:28:09.07,Default,,0000,0000,0000,,Now, even if we've never met 125\Nbefore, in these terms, one of
Dialogue: 0,0:28:09.07,0:28:15.61,Default,,0000,0000,0000,,things we ought to be aware of\Nis it ends in a 5, so it must
Dialogue: 0,0:28:15.61,0:28:20.52,Default,,0000,0000,0000,,divide by 5. So it's a fair\Nguess that 5 multiplied by
Dialogue: 0,0:28:20.52,0:28:25.84,Default,,0000,0000,0000,,itself three times would give us\N125, and indeed it does so to
Dialogue: 0,0:28:25.84,0:28:30.74,Default,,0000,0000,0000,,the third power, 125 to the\Nthird power is 5. The number
Dialogue: 0,0:28:30.74,0:28:31.97,Default,,0000,0000,0000,,multiplied by itself.
Dialogue: 0,0:28:32.64,0:28:39.20,Default,,0000,0000,0000,,Three times that gives us 125 is\N5 and now we just want to square
Dialogue: 0,0:28:39.20,0:28:41.82,Default,,0000,0000,0000,,that and that obviously gives us
Dialogue: 0,0:28:41.82,0:28:49.58,Default,,0000,0000,0000,,25. Take 8 - 2/3\Nagain. Let's deal with the minus
Dialogue: 0,0:28:49.58,0:28:55.30,Default,,0000,0000,0000,,sign. That is one over 8\Nto the 2/3.
Dialogue: 0,0:28:55.86,0:29:01.35,Default,,0000,0000,0000,,Again, let's deal with the\Nthird bit, the root bit first.
Dialogue: 0,0:29:01.35,0:29:05.84,Default,,0000,0000,0000,,What number, when multiplied\Nby itself three times, would
Dialogue: 0,0:29:05.84,0:29:12.83,Default,,0000,0000,0000,,give us eight that must be 2\NAnn. We need to square it and
Dialogue: 0,0:29:12.83,0:29:14.82,Default,,0000,0000,0000,,so that becomes 1/4.
Dialogue: 0,0:29:18.49,0:29:24.77,Default,,0000,0000,0000,,We take\None over
Dialogue: 0,0:29:24.77,0:29:31.06,Default,,0000,0000,0000,,25 to\Nthe minus
Dialogue: 0,0:29:31.06,0:29:38.71,Default,,0000,0000,0000,,2. Well, remember we\Nsaw what this happened with the
Dialogue: 0,0:29:38.71,0:29:46.06,Default,,0000,0000,0000,,one over 7 to the minus\Ntwo this became 25 squared and
Dialogue: 0,0:29:46.06,0:29:48.52,Default,,0000,0000,0000,,that is simply 625.
Dialogue: 0,0:29:50.19,0:29:53.61,Default,,0000,0000,0000,,243 to
Dialogue: 0,0:29:53.61,0:30:01.55,Default,,0000,0000,0000,,the 3/5.\NLet's do the 5th bit 1st
Dialogue: 0,0:30:01.55,0:30:07.90,Default,,0000,0000,0000,,and then the QB. Well 243,\Nthat's three multiplied by
Dialogue: 0,0:30:07.90,0:30:15.52,Default,,0000,0000,0000,,itself five times. So that is\Na three race to the power
Dialogue: 0,0:30:15.52,0:30:18.06,Default,,0000,0000,0000,,3, and that's 27.
Dialogue: 0,0:30:19.39,0:30:26.64,Default,,0000,0000,0000,,One last example,\Nlet's take 81
Dialogue: 0,0:30:26.64,0:30:30.27,Default,,0000,0000,0000,,and to the
Dialogue: 0,0:30:30.27,0:30:37.17,Default,,0000,0000,0000,,minus 3/4.\NOK, minus sign means
Dialogue: 0,0:30:37.17,0:30:43.99,Default,,0000,0000,0000,,one over. Where dividing\Nby a fraction to
Dialogue: 0,0:30:43.99,0:30:50.73,Default,,0000,0000,0000,,divide by a fraction,\Nwe invert and multiply,
Dialogue: 0,0:30:50.73,0:30:57.46,Default,,0000,0000,0000,,so this becomes 16\Nover 81 raised to
Dialogue: 0,0:30:57.46,0:30:59.99,Default,,0000,0000,0000,,the power 3/4.
Dialogue: 0,0:31:00.05,0:31:07.05,Default,,0000,0000,0000,,Equals 16 over 81.\NFirst of all to
Dialogue: 0,0:31:07.05,0:31:10.55,Default,,0000,0000,0000,,the quarter and then.
Dialogue: 0,0:31:10.56,0:31:15.82,Default,,0000,0000,0000,,Raised to the power three to the\Nquarter, this is 2 to the power
Dialogue: 0,0:31:15.82,0:31:21.09,Default,,0000,0000,0000,,four on top 16 is 2 to the power\Nfour. So that's a 2.
Dialogue: 0,0:31:21.86,0:31:28.55,Default,,0000,0000,0000,,And 81 is 3 to the power four,\Nso that's a 3 and we need to
Dialogue: 0,0:31:28.55,0:31:32.31,Default,,0000,0000,0000,,cube that and that gives us\Neight over 27th.
Dialogue: 0,0:31:33.13,0:31:37.24,Default,,0000,0000,0000,,So working with these indices\Nshouldn't really be too
Dialogue: 0,0:31:37.24,0:31:42.27,Default,,0000,0000,0000,,difficult. You need to remember\Nhow numbers are made up. You
Dialogue: 0,0:31:42.27,0:31:48.21,Default,,0000,0000,0000,,need to have in your mind's eye\Nthat 243 is 3 multiplied by
Dialogue: 0,0:31:48.21,0:31:54.15,Default,,0000,0000,0000,,itself. 5 * 243 is 3 to the\Npower five, and similarly with
Dialogue: 0,0:31:54.15,0:31:56.44,Default,,0000,0000,0000,,numbers like 30, two, 64128.
Dialogue: 0,0:31:57.75,0:32:02.96,Default,,0000,0000,0000,,If you can carry those around in\Nyour head and you will do with
Dialogue: 0,0:32:02.96,0:32:07.42,Default,,0000,0000,0000,,practice, you get to know them\Nquite well. Then this kind of
Dialogue: 0,0:32:07.42,0:32:11.14,Default,,0000,0000,0000,,calculation should become very\Neasy to remember. Deal with the
Dialogue: 0,0:32:11.14,0:32:15.61,Default,,0000,0000,0000,,minus sign first. By and large.\NGet that to be positive, then
Dialogue: 0,0:32:15.61,0:32:19.70,Default,,0000,0000,0000,,deal with the root sign that\Ngets the number. Work down.
Dialogue: 0,0:32:19.70,0:32:20.81,Default,,0000,0000,0000,,Keeps it small.
Dialogue: 0,0:32:21.74,0:32:24.44,Default,,0000,0000,0000,,They can be tricky, but we\Nhope we've shown that they
Dialogue: 0,0:32:24.44,0:32:25.17,Default,,0000,0000,0000,,can be mastered.