[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.75,0:00:04.66,Default,,0000,0000,0000,,Today we're going to look at\Nnumbers, written his powers, and
Dialogue: 0,0:00:04.66,0:00:08.20,Default,,0000,0000,0000,,we're going to go through some\Ncalculations involving them, and
Dialogue: 0,0:00:08.20,0:00:11.40,Default,,0000,0000,0000,,in particular, we're going to\Nlook at square roots.
Dialogue: 0,0:00:12.18,0:00:16.45,Default,,0000,0000,0000,,And the square roots of numbers,\Nwhich give us an irrational
Dialogue: 0,0:00:16.45,0:00:18.25,Default,,0000,0000,0000,,answer. That is.
Dialogue: 0,0:00:18.80,0:00:23.12,Default,,0000,0000,0000,,Numbers which can be written as\Nwhole numbers are fractions. Now
Dialogue: 0,0:00:23.12,0:00:25.48,Default,,0000,0000,0000,,these types of square roots are
Dialogue: 0,0:00:25.48,0:00:28.07,Default,,0000,0000,0000,,called Surds. But more of that
Dialogue: 0,0:00:28.07,0:00:31.83,Default,,0000,0000,0000,,later. What I want to do first\Nis a little bit of revision.
Dialogue: 0,0:00:32.71,0:00:35.72,Default,,0000,0000,0000,,And we'll start with something\Nthat we know.
Dialogue: 0,0:00:36.81,0:00:43.51,Default,,0000,0000,0000,,2 cubed Is written as\N2 * 2 * 2.
Dialogue: 0,0:00:45.00,0:00:48.69,Default,,0000,0000,0000,,It's a neat way of writing this
Dialogue: 0,0:00:48.69,0:00:55.05,Default,,0000,0000,0000,,repeated multiplication. And we\Nsay that three is the power or
Dialogue: 0,0:00:55.05,0:01:00.23,Default,,0000,0000,0000,,index. And two is rears to that\Npower or index.
Dialogue: 0,0:01:01.72,0:01:06.62,Default,,0000,0000,0000,,And the value of 2 cubed\Nworks out to be it because
Dialogue: 0,0:01:06.62,0:01:09.06,Default,,0000,0000,0000,,it's 2 * 2 * 2.
Dialogue: 0,0:01:10.45,0:01:14.41,Default,,0000,0000,0000,,So if we had 4 cubed.
Dialogue: 0,0:01:15.43,0:01:19.35,Default,,0000,0000,0000,,That is 4 * 4
Dialogue: 0,0:01:19.35,0:01:24.12,Default,,0000,0000,0000,,* 4. And we know when that's\Nworth. Diet is 64.
Dialogue: 0,0:01:24.84,0:01:27.51,Default,,0000,0000,0000,,But what if we have 4 to the
Dialogue: 0,0:01:27.51,0:01:32.75,Default,,0000,0000,0000,,negative 3? What would be its\Nvalue and high? Could we put it
Dialogue: 0,0:01:32.75,0:01:33.78,Default,,0000,0000,0000,,down more deeply?
Dialogue: 0,0:01:34.40,0:01:36.37,Default,,0000,0000,0000,,Well, we go back to what we
Dialogue: 0,0:01:36.37,0:01:42.36,Default,,0000,0000,0000,,know. 4 cubed equals 4 * 4\N* 4, which is 64.
Dialogue: 0,0:01:43.22,0:01:49.79,Default,,0000,0000,0000,,4 squared equals 4\N* 4 and not
Dialogue: 0,0:01:49.79,0:01:56.86,Default,,0000,0000,0000,,equals 16. 14\Npower one is for.
Dialogue: 0,0:01:56.86,0:01:58.52,Default,,0000,0000,0000,,Is for.
Dialogue: 0,0:01:59.64,0:02:05.45,Default,,0000,0000,0000,,An forwarded the zero. Well, if\Nwe look at our pattern in our
Dialogue: 0,0:02:05.45,0:02:09.97,Default,,0000,0000,0000,,calculations. We are dividing by\Nfor each time.
Dialogue: 0,0:02:10.57,0:02:16.78,Default,,0000,0000,0000,,And the index. Our power\Ndecreases by one. So forward the
Dialogue: 0,0:02:16.78,0:02:20.74,Default,,0000,0000,0000,,zero must be 4 / 4 which
Dialogue: 0,0:02:20.74,0:02:27.22,Default,,0000,0000,0000,,equals 1. But we were interested\Nin finding out about four to the
Dialogue: 0,0:02:27.22,0:02:30.82,Default,,0000,0000,0000,,negative three, so will continue\Nour pattern of multiplications.
Dialogue: 0,0:02:31.33,0:02:38.97,Default,,0000,0000,0000,,4 to the negative one will\Nbe the 1 / 4 which
Dialogue: 0,0:02:38.97,0:02:45.66,Default,,0000,0000,0000,,is 1/4. And forward to\Nthe negative two must be 1/4.
Dialogue: 0,0:02:46.22,0:02:49.79,Default,,0000,0000,0000,,Divided by 4, which is 16.
Dialogue: 0,0:02:50.61,0:02:57.76,Default,,0000,0000,0000,,An four to the negative three\Nmust be 116th divided by 4,
Dialogue: 0,0:02:57.76,0:03:00.74,Default,,0000,0000,0000,,which is one over 64.
Dialogue: 0,0:03:02.05,0:03:09.09,Default,,0000,0000,0000,,But If we go back to this\Nline, we knew that 4 cubed is
Dialogue: 0,0:03:09.09,0:03:14.83,Default,,0000,0000,0000,,64, so we can rewrite one over\N64 as one over 4 cubed.
Dialogue: 0,0:03:15.69,0:03:19.52,Default,,0000,0000,0000,,And look at that. Forward\Nto the negative three is
Dialogue: 0,0:03:19.52,0:03:21.02,Default,,0000,0000,0000,,one over 4 cubed.
Dialogue: 0,0:03:23.06,0:03:29.95,Default,,0000,0000,0000,,So for the negative two is one\Nover 16 must be one over 4
Dialogue: 0,0:03:29.95,0:03:34.96,Default,,0000,0000,0000,,squared. And four to the\Nnegative, one must equal
Dialogue: 0,0:03:34.96,0:03:39.87,Default,,0000,0000,0000,,for one over 4th power,\None which is one over 4.
Dialogue: 0,0:03:41.63,0:03:47.14,Default,,0000,0000,0000,,So negative powers give you the\Nreciprocal of the number.
Dialogue: 0,0:03:47.81,0:03:50.46,Default,,0000,0000,0000,,That is one over the number.
Dialogue: 0,0:03:51.31,0:03:55.07,Default,,0000,0000,0000,,So if I had this example.
Dialogue: 0,0:03:56.65,0:03:59.17,Default,,0000,0000,0000,,3 to the negative 2.
Dialogue: 0,0:03:59.76,0:04:04.17,Default,,0000,0000,0000,,We can write that as\None over 3 squared.
Dialogue: 0,0:04:06.05,0:04:09.78,Default,,0000,0000,0000,,And one over 3 squared is the\Nsame as one over 9.
Dialogue: 0,0:04:11.26,0:04:16.51,Default,,0000,0000,0000,,What about this one? 5 to the\Nnegative 3? How can we rewrite
Dialogue: 0,0:04:16.51,0:04:21.36,Default,,0000,0000,0000,,that and what's the value? Well,\Nthe negative 3 means it's one
Dialogue: 0,0:04:21.36,0:04:24.27,Default,,0000,0000,0000,,over. 5 cubed
Dialogue: 0,0:04:25.15,0:04:32.21,Default,,0000,0000,0000,,And that is one over 5 cubed\Nis 125. So 5 to the negative
Dialogue: 0,0:04:32.21,0:04:34.73,Default,,0000,0000,0000,,three is one over 125.
Dialogue: 0,0:04:36.95,0:04:41.93,Default,,0000,0000,0000,,Not the big thing to note is\Nthat even though you've got a
Dialogue: 0,0:04:41.93,0:04:46.14,Default,,0000,0000,0000,,negative power, that value that\Nyou get is a positive answer.
Dialogue: 0,0:04:46.14,0:04:51.12,Default,,0000,0000,0000,,That's a big mistake by quite a\Nlot of people. They think when
Dialogue: 0,0:04:51.12,0:04:55.33,Default,,0000,0000,0000,,you've got a negative power,\Nyou'll get a negative answer, so
Dialogue: 0,0:04:55.33,0:04:58.78,Default,,0000,0000,0000,,please don't make that mistake.\NAnd negative power means
Dialogue: 0,0:04:58.78,0:05:01.84,Default,,0000,0000,0000,,reciprocal of the number one\Nover the power.
Dialogue: 0,0:05:03.02,0:05:08.92,Default,,0000,0000,0000,,Moving on in the examples that\NI've just been doing, we've used
Dialogue: 0,0:05:08.92,0:05:14.34,Default,,0000,0000,0000,,positive negative integers for\Nthe powers, but what if we have
Dialogue: 0,0:05:14.34,0:05:16.30,Default,,0000,0000,0000,,fractional indices or fractional
Dialogue: 0,0:05:16.30,0:05:20.11,Default,,0000,0000,0000,,powers? How can we represent\Nthem? What's their value?
Dialogue: 0,0:05:20.75,0:05:22.11,Default,,0000,0000,0000,,Well again will start with what
Dialogue: 0,0:05:22.11,0:05:27.85,Default,,0000,0000,0000,,we know. We know that force of\Npower one is 4.
Dialogue: 0,0:05:28.46,0:05:33.68,Default,,0000,0000,0000,,But what would be for to the\Nhalf? What's it value?
Dialogue: 0,0:05:34.19,0:05:37.78,Default,,0000,0000,0000,,We're looking at this, we
Dialogue: 0,0:05:37.78,0:05:44.33,Default,,0000,0000,0000,,know using. Indices laws,\Ntherefore, to the half times by
Dialogue: 0,0:05:44.33,0:05:48.02,Default,,0000,0000,0000,,4 to the half, must equal 4.
Dialogue: 0,0:05:48.70,0:05:50.91,Default,,0000,0000,0000,,To the indices added together.
Dialogue: 0,0:05:51.50,0:05:53.45,Default,,0000,0000,0000,,And 1/2 + 1/2 is one.
Dialogue: 0,0:05:54.46,0:05:58.32,Default,,0000,0000,0000,,Before to the one is just 4.
Dialogue: 0,0:05:59.42,0:06:03.43,Default,,0000,0000,0000,,So 4 to the half times, 4\N1/2 equals 4.
Dialogue: 0,0:06:04.79,0:06:08.86,Default,,0000,0000,0000,,That means that it's something\Ntimes by itself equals 4. Well,
Dialogue: 0,0:06:08.86,0:06:13.30,Default,,0000,0000,0000,,we know that it's obvious it's\Ntwo 2 * 2 equals 4.
Dialogue: 0,0:06:14.25,0:06:17.27,Default,,0000,0000,0000,,So 4 to the half equals 2.
Dialogue: 0,0:06:17.78,0:06:21.28,Default,,0000,0000,0000,,So. Forza half.
Dialogue: 0,0:06:21.88,0:06:23.12,Default,,0000,0000,0000,,Is equal to 2.
Dialogue: 0,0:06:23.87,0:06:28.72,Default,,0000,0000,0000,,And we write this in a shorthand\Nnotation by doing this for the
Dialogue: 0,0:06:28.72,0:06:30.96,Default,,0000,0000,0000,,half is equal to the square
Dialogue: 0,0:06:30.96,0:06:36.42,Default,,0000,0000,0000,,root. Of four, and that is the\Nsymbol for the square root.
Dialogue: 0,0:06:36.93,0:06:41.58,Default,,0000,0000,0000,,So the half is associated\Nwith the square routing of a
Dialogue: 0,0:06:41.58,0:06:47.50,Default,,0000,0000,0000,,number. So if we had 9 to the\Npower of 1/2, that's the same
Dialogue: 0,0:06:47.50,0:06:51.31,Default,,0000,0000,0000,,as the square root of 9,\Nwhich equals 3.
Dialogue: 0,0:06:52.44,0:06:58.25,Default,,0000,0000,0000,,So in general, if we had a\Nnumber a razor, the part half.
Dialogue: 0,0:06:58.81,0:07:02.55,Default,,0000,0000,0000,,Then we can write that as the\Nsquare root of A.
Dialogue: 0,0:07:03.86,0:07:08.80,Default,,0000,0000,0000,,We can continue this on by\Nsaying if we wanted the cube
Dialogue: 0,0:07:08.80,0:07:11.28,Default,,0000,0000,0000,,root of a number, say for.
Dialogue: 0,0:07:12.26,0:07:18.04,Default,,0000,0000,0000,,To the third, we can write that\Nas the cube root of 4.
Dialogue: 0,0:07:19.44,0:07:20.23,Default,,0000,0000,0000,,Like this?
Dialogue: 0,0:07:21.34,0:07:26.87,Default,,0000,0000,0000,,And the fraction associated with\Nthe cube root is 1/3. Similarly,
Dialogue: 0,0:07:26.87,0:07:33.92,Default,,0000,0000,0000,,the 4th root. So if we had\Nfive, the 4th root of 5, you
Dialogue: 0,0:07:33.92,0:07:38.94,Default,,0000,0000,0000,,can write that using this\Nshorthand notation with a small
Dialogue: 0,0:07:38.94,0:07:44.48,Default,,0000,0000,0000,,for their and then this root\Nsymbol. And the quarter is
Dialogue: 0,0:07:44.48,0:07:46.99,Default,,0000,0000,0000,,associated with the 4th root.
Dialogue: 0,0:07:48.39,0:07:50.62,Default,,0000,0000,0000,,So if we had the NTH root.
Dialogue: 0,0:07:51.17,0:07:53.86,Default,,0000,0000,0000,,Of a number A.
Dialogue: 0,0:07:54.38,0:07:59.08,Default,,0000,0000,0000,,We can write it like\Nthis in shorthand.
Dialogue: 0,0:08:01.94,0:08:05.50,Default,,0000,0000,0000,,So if we have the\Ncube root of 64.
Dialogue: 0,0:08:06.73,0:08:09.80,Default,,0000,0000,0000,,We can write it like this.
Dialogue: 0,0:08:10.54,0:08:17.58,Default,,0000,0000,0000,,And it equals 64. Raise\Nthe part of 1/3. Now
Dialogue: 0,0:08:17.58,0:08:23.92,Default,,0000,0000,0000,,we know 64 is 4\N* 4 * 4.
Dialogue: 0,0:08:24.45,0:08:28.21,Default,,0000,0000,0000,,And that's raised to the power
Dialogue: 0,0:08:28.21,0:08:33.11,Default,,0000,0000,0000,,of 1/3. And the answer is dead\Neasy. We want to know what
Dialogue: 0,0:08:33.11,0:08:36.76,Default,,0000,0000,0000,,number times by itself three\Ntimes gives you 64. Well, it's
Dialogue: 0,0:08:36.76,0:08:40.75,Default,,0000,0000,0000,,obvious because what we've just\Nwritten it must be for. So the
Dialogue: 0,0:08:40.75,0:08:42.74,Default,,0000,0000,0000,,cube root of 64 is 4.
Dialogue: 0,0:08:44.92,0:08:50.24,Default,,0000,0000,0000,,A very useful simple fact\Nleading on from this that.
Dialogue: 0,0:08:51.10,0:08:56.65,Default,,0000,0000,0000,,Is known, but people tend to\Nforget it is that we know that
Dialogue: 0,0:08:56.65,0:09:01.78,Default,,0000,0000,0000,,forward to the half times by 4\Nto the half equals 4.
Dialogue: 0,0:09:02.89,0:09:09.58,Default,,0000,0000,0000,,That is the square root of 4\Ntimes by the square root of 4 is
Dialogue: 0,0:09:09.58,0:09:15.38,Default,,0000,0000,0000,,4 and we can write that more\Nsimply by saying is the square
Dialogue: 0,0:09:15.38,0:09:18.05,Default,,0000,0000,0000,,root of 4 squared equals 4.
Dialogue: 0,0:09:19.24,0:09:24.27,Default,,0000,0000,0000,,Not, I'll be using this very\Nsimple fact. Little role at the
Dialogue: 0,0:09:24.27,0:09:29.30,Default,,0000,0000,0000,,end of the session, so remember\Nit. It's any number. The square
Dialogue: 0,0:09:29.30,0:09:31.81,Default,,0000,0000,0000,,root of it squared gives you
Dialogue: 0,0:09:31.81,0:09:37.41,Default,,0000,0000,0000,,that number. Another important\Nthing to remember about serves
Dialogue: 0,0:09:37.41,0:09:43.44,Default,,0000,0000,0000,,comes from solving this\Nequation. If we had this very
Dialogue: 0,0:09:43.44,0:09:47.66,Default,,0000,0000,0000,,simple quadratic equation X\Nsquared equal 4.
Dialogue: 0,0:09:48.46,0:09:54.31,Default,,0000,0000,0000,,If we take the square root of\Nfour, we know that we have to
Dialogue: 0,0:09:54.31,0:09:55.57,Default,,0000,0000,0000,,have two routes.
Dialogue: 0,0:09:56.10,0:10:01.04,Default,,0000,0000,0000,,For the solution, because this\Nis a quadratic equation, so X
Dialogue: 0,0:10:01.04,0:10:05.08,Default,,0000,0000,0000,,equals plus or minus the square\Nroot of 4.
Dialogue: 0,0:10:05.64,0:10:09.46,Default,,0000,0000,0000,,So we know that is\Npositive two or
Dialogue: 0,0:10:09.46,0:10:12.33,Default,,0000,0000,0000,,negative two. You've\Ngot 2 routes.
Dialogue: 0,0:10:13.71,0:10:17.60,Default,,0000,0000,0000,,So. Roots are not unique.
Dialogue: 0,0:10:18.25,0:10:22.35,Default,,0000,0000,0000,,And remember, we don't\Nalways have to write the
Dialogue: 0,0:10:22.35,0:10:26.46,Default,,0000,0000,0000,,positive side. We can write\Npositive two as two.
Dialogue: 0,0:10:27.65,0:10:32.82,Default,,0000,0000,0000,,And remember. If you haven't got\Na sign in front of the square
Dialogue: 0,0:10:32.82,0:10:34.71,Default,,0000,0000,0000,,root, then it's assumed it's a
Dialogue: 0,0:10:34.71,0:10:39.37,Default,,0000,0000,0000,,positive square root. When it's\Nnot, you have to put the sign in
Dialogue: 0,0:10:39.37,0:10:41.03,Default,,0000,0000,0000,,and make it a negative square
Dialogue: 0,0:10:41.03,0:10:44.63,Default,,0000,0000,0000,,root. But the positive\Nsquare roots of the ones
Dialogue: 0,0:10:44.63,0:10:46.08,Default,,0000,0000,0000,,that are are most common.
Dialogue: 0,0:10:48.30,0:10:49.72,Default,,0000,0000,0000,,Now moving on.
Dialogue: 0,0:10:50.28,0:10:52.27,Default,,0000,0000,0000,,What I'd like to do is actually
Dialogue: 0,0:10:52.27,0:10:54.60,Default,,0000,0000,0000,,do some work. On
Dialogue: 0,0:10:55.78,0:10:57.98,Default,,0000,0000,0000,,The square root of negative
Dialogue: 0,0:10:57.98,0:11:03.06,Default,,0000,0000,0000,,numbers. We've done the square\Nroot of positive numbers. What
Dialogue: 0,0:11:03.06,0:11:06.08,Default,,0000,0000,0000,,if we had the square root of
Dialogue: 0,0:11:06.08,0:11:09.12,Default,,0000,0000,0000,,negative 9? Can we evaluate
Dialogue: 0,0:11:09.12,0:11:13.99,Default,,0000,0000,0000,,this? Well, the square root of\Nnegative 9 can be written as
Dialogue: 0,0:11:13.99,0:11:15.89,Default,,0000,0000,0000,,negative 9 raised to the power
Dialogue: 0,0:11:15.89,0:11:21.28,Default,,0000,0000,0000,,of 1/2. And So what we're\Nlooking for are two numbers
Dialogue: 0,0:11:21.28,0:11:25.48,Default,,0000,0000,0000,,multiplied by themselves, which\Nwill give us negative. Now when
Dialogue: 0,0:11:25.48,0:11:28.42,Default,,0000,0000,0000,,we know 3 * 3 is 9.
Dialogue: 0,0:11:29.27,0:11:33.17,Default,,0000,0000,0000,,And negative three times\Nnegative three is 9.
Dialogue: 0,0:11:34.41,0:11:35.43,Default,,0000,0000,0000,,But there's not.
Dialogue: 0,0:11:36.12,0:11:39.40,Default,,0000,0000,0000,,2 numbers multiply by\Nthemselves, which will give us
Dialogue: 0,0:11:39.40,0:11:42.66,Default,,0000,0000,0000,,negative night. So you\Ncannot evaluate the square
Dialogue: 0,0:11:42.66,0:11:45.84,Default,,0000,0000,0000,,root of a negative number\Nand get a real answer.
Dialogue: 0,0:11:48.20,0:11:55.38,Default,,0000,0000,0000,,Now what I'd like to do is\Ncontinue on using surds in some
Dialogue: 0,0:11:55.38,0:12:00.41,Default,,0000,0000,0000,,calculations. And go over some\Ncommon square roots that you
Dialogue: 0,0:12:00.41,0:12:07.93,Default,,0000,0000,0000,,might know. Now we all know\Nthat the square root of 25 is 5
Dialogue: 0,0:12:07.93,0:12:10.86,Default,,0000,0000,0000,,because 5 * 5 is 25.
Dialogue: 0,0:12:11.47,0:12:15.91,Default,,0000,0000,0000,,And if we were given the\Nsquare root of 9 over 4,
Dialogue: 0,0:12:15.91,0:12:21.09,Default,,0000,0000,0000,,that's dead easy. It must be\N3 over 2 because 3 * 3 is
Dialogue: 0,0:12:21.09,0:12:26.64,Default,,0000,0000,0000,,nine, 2 * 2 is 4, and we\Nwrite that as one and a half.
Dialogue: 0,0:12:28.54,0:12:32.36,Default,,0000,0000,0000,,But we've now taken square root\Nof whole numbers and fractions,
Dialogue: 0,0:12:32.36,0:12:35.83,Default,,0000,0000,0000,,and we've got answers that are\Nwhole numbers and fractions.
Dialogue: 0,0:12:36.36,0:12:41.62,Default,,0000,0000,0000,,But there are other square roots\Nin which we don't get whole
Dialogue: 0,0:12:41.62,0:12:45.56,Default,,0000,0000,0000,,numbers or fractions. We get\Nthem as non terminating
Dialogue: 0,0:12:45.56,0:12:48.88,Default,,0000,0000,0000,,decimals. Such as Route 2.
Dialogue: 0,0:12:49.41,0:12:55.64,Default,,0000,0000,0000,,Route 2 cannot be evaluated as a\Nwhole number or a fraction.
Dialogue: 0,0:12:55.64,0:12:57.71,Default,,0000,0000,0000,,Never can Route 3.
Dialogue: 0,0:12:59.24,0:13:04.10,Default,,0000,0000,0000,,We can only write them\Nas approximations. They
Dialogue: 0,0:13:04.10,0:13:05.93,Default,,0000,0000,0000,,are irrational numbers.
Dialogue: 0,0:13:07.04,0:13:13.34,Default,,0000,0000,0000,,An approximation for the square\Nroot of 2 to 3 decimal places
Dialogue: 0,0:13:13.34,0:13:19.70,Default,,0000,0000,0000,,is 1.414. Approximation from\NRoute 3 to 3 decimal places,
Dialogue: 0,0:13:19.70,0:13:26.86,Default,,0000,0000,0000,,and I put 3 decimal places\Njust to remind us is 1.732.
Dialogue: 0,0:13:27.78,0:13:33.10,Default,,0000,0000,0000,,Not because. They cannot be\Nevaluated as whole numbers
Dialogue: 0,0:13:33.10,0:13:36.34,Default,,0000,0000,0000,,and fractions, and these are\Nirrational numbers.
Dialogue: 0,0:13:37.91,0:13:41.97,Default,,0000,0000,0000,,They are very special and we\Ncall them surged.
Dialogue: 0,0:13:42.55,0:13:50.31,Default,,0000,0000,0000,,Other common starts\Nwould be Route
Dialogue: 0,0:13:50.31,0:13:57.99,Default,,0000,0000,0000,,5. Rich sex, which 7\NRoute 8 not Route 9 because we
Dialogue: 0,0:13:57.99,0:14:01.40,Default,,0000,0000,0000,,know that's three and Route 10.
Dialogue: 0,0:14:02.73,0:14:04.28,Default,,0000,0000,0000,,Now, why do we need to know
Dialogue: 0,0:14:04.28,0:14:08.94,Default,,0000,0000,0000,,about search? Will search crop\Nup a lot when we using
Dialogue: 0,0:14:08.94,0:14:12.98,Default,,0000,0000,0000,,Pythagoras Theorem and in\Ntrigonometry and so we need to
Dialogue: 0,0:14:12.98,0:14:15.42,Default,,0000,0000,0000,,know how to manipulate these in
Dialogue: 0,0:14:15.42,0:14:19.36,Default,,0000,0000,0000,,various calculations. And we\Nneed to know how to simplify
Dialogue: 0,0:14:19.36,0:14:20.19,Default,,0000,0000,0000,,them as well.
Dialogue: 0,0:14:21.36,0:14:26.52,Default,,0000,0000,0000,,Now Route 2 cannot be simplified\Nany further. Now they can Route
Dialogue: 0,0:14:26.52,0:14:33.40,Default,,0000,0000,0000,,3 or Route 5 with six, 3, Seven,\Nbut look at root it. If we take
Dialogue: 0,0:14:33.40,0:14:35.55,Default,,0000,0000,0000,,the square root of it.
Dialogue: 0,0:14:37.61,0:14:42.38,Default,,0000,0000,0000,,It can be replaced by the\Nproduct 4 * 2.
Dialogue: 0,0:14:43.30,0:14:46.79,Default,,0000,0000,0000,,Now the square root of product.
Dialogue: 0,0:14:47.60,0:14:52.70,Default,,0000,0000,0000,,Is the product of the square\Nroots, so it's Square root of 4
Dialogue: 0,0:14:52.70,0:14:55.44,Default,,0000,0000,0000,,times by the square root of 2.
Dialogue: 0,0:14:56.58,0:14:59.44,Default,,0000,0000,0000,,We know the square root of 4 is
Dialogue: 0,0:14:59.44,0:15:05.33,Default,,0000,0000,0000,,2. And this is Route 2. So the\Nsquare root of it can be written
Dialogue: 0,0:15:05.33,0:15:09.59,Default,,0000,0000,0000,,more simply as two square root\Nof two or two Route 2.
Dialogue: 0,0:15:10.16,0:15:16.44,Default,,0000,0000,0000,,In shorthand. What about this\None? What about the square root
Dialogue: 0,0:15:16.44,0:15:19.95,Default,,0000,0000,0000,,of 5 times by the square root
Dialogue: 0,0:15:19.95,0:15:25.68,Default,,0000,0000,0000,,of 15? Now there in the previous\Nexample, the product of the
Dialogue: 0,0:15:25.68,0:15:31.34,Default,,0000,0000,0000,,square roots is equal to the\Nsquare root of the products, so
Dialogue: 0,0:15:31.34,0:15:33.70,Default,,0000,0000,0000,,we use that fact too.
Dialogue: 0,0:15:34.27,0:15:36.27,Default,,0000,0000,0000,,Rewrite this expression in
Dialogue: 0,0:15:36.27,0:15:42.08,Default,,0000,0000,0000,,another way. The square root of\N5 times by the square root of 15
Dialogue: 0,0:15:42.08,0:15:44.89,Default,,0000,0000,0000,,is the square root of 5 times by
Dialogue: 0,0:15:44.89,0:15:49.83,Default,,0000,0000,0000,,15. And that is the square\Nroot of 75.
Dialogue: 0,0:15:50.45,0:15:58.26,Default,,0000,0000,0000,,But 75 can be rewritten\Nas 25 times by three.
Dialogue: 0,0:15:59.61,0:16:05.42,Default,,0000,0000,0000,,Note 25 is a square number and\Nwe can then separate that out to
Dialogue: 0,0:16:05.42,0:16:11.64,Default,,0000,0000,0000,,say that is the square root of\N25 times by the square root of 3
Dialogue: 0,0:16:11.64,0:16:18.28,Default,,0000,0000,0000,,square root of 25 is 5 times by\NRoute 3. So the square root of 5
Dialogue: 0,0:16:18.28,0:16:24.10,Default,,0000,0000,0000,,times the square root of 15 is 5\Nroute 3, but could be easier.
Dialogue: 0,0:16:24.97,0:16:31.70,Default,,0000,0000,0000,,But a common mistake is\Nthe people would take an
Dialogue: 0,0:16:31.70,0:16:33.72,Default,,0000,0000,0000,,expression like this.
Dialogue: 0,0:16:34.56,0:16:39.25,Default,,0000,0000,0000,,I served in the form of the\Nsquare root of 4 + 9 and say is
Dialogue: 0,0:16:39.25,0:16:43.06,Default,,0000,0000,0000,,equal to the square root of 4\Nplus the square root of 9.
Dialogue: 0,0:16:43.64,0:16:47.02,Default,,0000,0000,0000,,Now this can't happen\Nbecause if we follow this
Dialogue: 0,0:16:47.02,0:16:51.16,Default,,0000,0000,0000,,through that would be 2 +\N3, which would equal 5.
Dialogue: 0,0:16:52.45,0:16:57.11,Default,,0000,0000,0000,,This is not correct because five\Nis the square root of 25.
Dialogue: 0,0:16:57.77,0:17:05.37,Default,,0000,0000,0000,,And here we've got 4 + 9, which\Nis 13. So the square root of an
Dialogue: 0,0:17:05.37,0:17:10.60,Default,,0000,0000,0000,,addition is not equal to the\Nsquare root plus the other
Dialogue: 0,0:17:10.60,0:17:12.97,Default,,0000,0000,0000,,square root. So Please remember
Dialogue: 0,0:17:12.97,0:17:18.97,Default,,0000,0000,0000,,that. Those two rules that we\Npreviously did with Surds, an
Dialogue: 0,0:17:18.97,0:17:22.39,Default,,0000,0000,0000,,expansion of multiplications\Nonly applies to multiplication
Dialogue: 0,0:17:22.39,0:17:25.80,Default,,0000,0000,0000,,and division, not to addition\Nand Subtraction.
Dialogue: 0,0:17:29.23,0:17:31.25,Default,,0000,0000,0000,,So if we move on.
Dialogue: 0,0:17:32.00,0:17:38.32,Default,,0000,0000,0000,,And we take the square root\Nof bigger numbers times by.
Dialogue: 0,0:17:38.33,0:17:41.35,Default,,0000,0000,0000,,Square root of a another\Nbig number.
Dialogue: 0,0:17:42.63,0:17:46.41,Default,,0000,0000,0000,,The square root of 400 times by\Nthe square root of 90.
Dialogue: 0,0:17:47.48,0:17:50.96,Default,,0000,0000,0000,,It's useful to remember\Nsquare numbers. The
Dialogue: 0,0:17:50.96,0:17:52.45,Default,,0000,0000,0000,,typical square numbers.
Dialogue: 0,0:17:53.72,0:18:00.06,Default,,0000,0000,0000,,Square numbers of the numbers 1\Nto 10 at one, four 916-2549,
Dialogue: 0,0:18:00.06,0:18:05.86,Default,,0000,0000,0000,,sixty, 481 and 100. You should\Nbe able to remember those
Dialogue: 0,0:18:05.86,0:18:11.67,Default,,0000,0000,0000,,straight off and recall them,\Nand this helps us to rewrite
Dialogue: 0,0:18:11.67,0:18:17.99,Default,,0000,0000,0000,,these. Square roots we can write\Nsquare root of 400 as the square
Dialogue: 0,0:18:17.99,0:18:20.61,Default,,0000,0000,0000,,root of 4 times by 100.
Dialogue: 0,0:18:21.31,0:18:26.24,Default,,0000,0000,0000,,And the square root of 90 is the\Nsquare root of 9 * 10.
Dialogue: 0,0:18:26.75,0:18:29.37,Default,,0000,0000,0000,,Then we expand out.
Dialogue: 0,0:18:30.08,0:18:34.88,Default,,0000,0000,0000,,And we say that this product and\Nthe square root of that product
Dialogue: 0,0:18:34.88,0:18:39.67,Default,,0000,0000,0000,,is the square root of 4 times by\Nthe square root of 100.
Dialogue: 0,0:18:40.22,0:18:45.23,Default,,0000,0000,0000,,Times by the square root of 9\Ntimes by the square root of 10.
Dialogue: 0,0:18:45.79,0:18:50.23,Default,,0000,0000,0000,,Then we simplify. We know the\Nsquare root of 4 because there's
Dialogue: 0,0:18:50.23,0:18:55.41,Default,,0000,0000,0000,,a square number and it's two. We\Nknow the square root of 100 is
Dialogue: 0,0:18:55.41,0:18:58.00,Default,,0000,0000,0000,,10 square root of 9 is 3.
Dialogue: 0,0:18:58.60,0:19:02.80,Default,,0000,0000,0000,,Thought the square root of 10 is\Njust the square root of 10.
Dialogue: 0,0:19:04.45,0:19:08.97,Default,,0000,0000,0000,,And we meeting it up by\Nmultiplying these three numbers
Dialogue: 0,0:19:08.97,0:19:14.39,Default,,0000,0000,0000,,together. That's 2 * 10 * 3 is\N6060 Route 10 delicious.
Dialogue: 0,0:19:15.00,0:19:18.07,Default,,0000,0000,0000,,What if we had a quotient?
Dialogue: 0,0:19:18.97,0:19:20.73,Default,,0000,0000,0000,,Involving square roots.
Dialogue: 0,0:19:21.29,0:19:26.22,Default,,0000,0000,0000,,What if we had the square root\Nof 2000 divided by the square
Dialogue: 0,0:19:26.22,0:19:31.52,Default,,0000,0000,0000,,root of 50? Well, we use a\Nsimilar rule to the way that we
Dialogue: 0,0:19:31.52,0:19:36.45,Default,,0000,0000,0000,,use the rule for multiplication.\NWe can say that that is equal to
Dialogue: 0,0:19:36.45,0:19:39.10,Default,,0000,0000,0000,,the square root of 2000 / 50.
Dialogue: 0,0:19:39.64,0:19:42.15,Default,,0000,0000,0000,,We cancelled I'm.
Dialogue: 0,0:19:42.76,0:19:44.47,Default,,0000,0000,0000,,And we get.
Dialogue: 0,0:19:45.30,0:19:51.68,Default,,0000,0000,0000,,That The answer is the square\Nroot of 40, but we can simplify
Dialogue: 0,0:19:51.68,0:19:56.24,Default,,0000,0000,0000,,the square root of 40. Thinking\Nof square numbers breakdown 40
Dialogue: 0,0:19:56.24,0:20:02.06,Default,,0000,0000,0000,,as 4 * 10 and that equals the\Nsquare root of four times the
Dialogue: 0,0:20:02.06,0:20:07.45,Default,,0000,0000,0000,,square root of 10, which is 2\Ntimes the square root of 10.
Dialogue: 0,0:20:07.45,0:20:11.18,Default,,0000,0000,0000,,What could be simpler? It\Nbecomes second nature really,
Dialogue: 0,0:20:11.18,0:20:15.75,Default,,0000,0000,0000,,the more practice you do with\Nthe rules, the more you
Dialogue: 0,0:20:15.75,0:20:16.100,Default,,0000,0000,0000,,understand them and.
Dialogue: 0,0:20:17.02,0:20:18.27,Default,,0000,0000,0000,,You don't have to think\Nabout them.
Dialogue: 0,0:20:19.85,0:20:25.25,Default,,0000,0000,0000,,Now, these simplifications of\NSurds are quite easy ones.
Dialogue: 0,0:20:25.99,0:20:30.22,Default,,0000,0000,0000,,But what if we had this\Nexpression involving Surds?
Dialogue: 0,0:20:30.76,0:20:36.25,Default,,0000,0000,0000,,It's a little bit different from\Nthe expressions that we've
Dialogue: 0,0:20:36.25,0:20:41.82,Default,,0000,0000,0000,,already used. One plus square\Nroot of 3 times by two minus the
Dialogue: 0,0:20:41.82,0:20:46.14,Default,,0000,0000,0000,,square root of 2. How do we\Nexpand out an expression like
Dialogue: 0,0:20:46.14,0:20:51.18,Default,,0000,0000,0000,,this where we do it just in the\Nsame way as we did normally?
Dialogue: 0,0:20:52.21,0:20:56.72,Default,,0000,0000,0000,,We use the first number in these\Nbrackets, multiply it by these
Dialogue: 0,0:20:56.72,0:21:01.23,Default,,0000,0000,0000,,two numbers, and then we use\Nthis number and multiply it by
Dialogue: 0,0:21:01.23,0:21:05.37,Default,,0000,0000,0000,,these two numbers and combine\Nthem together if possible. So if
Dialogue: 0,0:21:05.37,0:21:10.63,Default,,0000,0000,0000,,we do that, we multiply 1 by\Nthese two numbers and we get 2
Dialogue: 0,0:21:10.63,0:21:15.52,Default,,0000,0000,0000,,minus the square root of 2 and\Nwe multiplied by square root of
Dialogue: 0,0:21:15.52,0:21:16.65,Default,,0000,0000,0000,,3 that is.
Dialogue: 0,0:21:17.41,0:21:24.16,Default,,0000,0000,0000,,Square root of 3 * 2 is 2 square\Nroot of 3 and then the square
Dialogue: 0,0:21:24.16,0:21:29.23,Default,,0000,0000,0000,,root of 3 times negative square\Nroot of 2, which is your
Dialogue: 0,0:21:29.23,0:21:32.71,Default,,0000,0000,0000,,subtract. Route 3 Route 2.
Dialogue: 0,0:21:33.71,0:21:35.56,Default,,0000,0000,0000,,Now when we look at that
Dialogue: 0,0:21:35.56,0:21:39.90,Default,,0000,0000,0000,,expression. And consider it you\Ncan see that we can't really
Dialogue: 0,0:21:39.90,0:21:41.31,Default,,0000,0000,0000,,simplify it any further.
Dialogue: 0,0:21:42.88,0:21:46.41,Default,,0000,0000,0000,,But what if we had this
Dialogue: 0,0:21:46.41,0:21:51.52,Default,,0000,0000,0000,,expression? Similar to the one\Nabove using two brackets, but
Dialogue: 0,0:21:51.52,0:21:56.40,Default,,0000,0000,0000,,look at the numbers and the\N3rd's involved. One plus square
Dialogue: 0,0:21:56.40,0:22:02.62,Default,,0000,0000,0000,,root of 3 times by one minus the\Nsquare root of 3. The numbers
Dialogue: 0,0:22:02.62,0:22:07.06,Default,,0000,0000,0000,,the same, but the operations\Ndifferent. See what happens when
Dialogue: 0,0:22:07.06,0:22:13.72,Default,,0000,0000,0000,,we multiply out in the usual\Nway. 1 * 1 minus Route 3 is 1
Dialogue: 0,0:22:13.72,0:22:19.49,Default,,0000,0000,0000,,minus Route 3, then Route 3\Ntimes these two Route 3 * 1.
Dialogue: 0,0:22:19.49,0:22:26.99,Default,,0000,0000,0000,,Is plus 3 route 3 times\Nby minus 3 three is subtract.
Dialogue: 0,0:22:26.99,0:22:29.49,Default,,0000,0000,0000,,Route 3 route 3.
Dialogue: 0,0:22:30.20,0:22:36.11,Default,,0000,0000,0000,,Now remember. That little fact\Nthat we did previously. This is
Dialogue: 0,0:22:36.11,0:22:42.21,Default,,0000,0000,0000,,Route 3 squared and what's Route\N3 squared? It's just three so we
Dialogue: 0,0:22:42.21,0:22:48.30,Default,,0000,0000,0000,,can look at our expression and\Nwe can see that that is one.
Dialogue: 0,0:22:48.87,0:22:51.39,Default,,0000,0000,0000,,And loan behold, look what\Nhappens here in the middle.
Dialogue: 0,0:22:52.67,0:22:55.72,Default,,0000,0000,0000,,Subtract right, three, add Route\N3, they cancel.
Dialogue: 0,0:22:56.30,0:22:59.32,Default,,0000,0000,0000,,And so we're left with one
Dialogue: 0,0:22:59.32,0:23:06.41,Default,,0000,0000,0000,,minus. This which is three\N1 - 3 is negative 2.
Dialogue: 0,0:23:08.38,0:23:12.35,Default,,0000,0000,0000,,So we started off with a\Nmultiplication of brackets,
Dialogue: 0,0:23:12.35,0:23:16.76,Default,,0000,0000,0000,,which involves surds. But look\Nwhat happens with the answer.
Dialogue: 0,0:23:16.76,0:23:20.73,Default,,0000,0000,0000,,We've got a whole number with\Nno service involved.
Dialogue: 0,0:23:22.13,0:23:27.11,Default,,0000,0000,0000,,Now this is a significant fact,\Nand in fact this is so well
Dialogue: 0,0:23:27.11,0:23:31.70,Default,,0000,0000,0000,,known it's given a name. This\Ntype of expansion is given a
Dialogue: 0,0:23:31.70,0:23:34.77,Default,,0000,0000,0000,,name. It's called the difference\Nof two squares.
Dialogue: 0,0:23:35.39,0:23:41.42,Default,,0000,0000,0000,,And the difference of two\Nsquares is written as a squared
Dialogue: 0,0:23:41.42,0:23:45.25,Default,,0000,0000,0000,,minus B squared, and it can be
Dialogue: 0,0:23:45.25,0:23:47.45,Default,,0000,0000,0000,,expanded. By using two brackets.
Dialogue: 0,0:23:48.63,0:23:52.40,Default,,0000,0000,0000,,In two A-B times
Dialogue: 0,0:23:52.40,0:23:58.91,Default,,0000,0000,0000,,by A+B? Now, can\Nyou see that this is similar to
Dialogue: 0,0:23:58.91,0:24:04.68,Default,,0000,0000,0000,,what we had up here, where a is\None and B is Route 3?
Dialogue: 0,0:24:05.84,0:24:13.54,Default,,0000,0000,0000,,So. Our product here. One\Nplus Route 3 times by one
Dialogue: 0,0:24:13.54,0:24:20.44,Default,,0000,0000,0000,,minus Route 3 is really the\Ndifference of two squares and
Dialogue: 0,0:24:20.44,0:24:24.20,Default,,0000,0000,0000,,the two squares are 1 squared.
Dialogue: 0,0:24:24.87,0:24:28.43,Default,,0000,0000,0000,,Minus Route 3 squared.
Dialogue: 0,0:24:29.36,0:24:34.92,Default,,0000,0000,0000,,Again, notice we're using that\Nsimple fact, which makes things
Dialogue: 0,0:24:34.92,0:24:41.04,Default,,0000,0000,0000,,really easy. That is 1 - 3,\Nwhich is negative 2.
Dialogue: 0,0:24:42.05,0:24:44.48,Default,,0000,0000,0000,,So if you remember this
Dialogue: 0,0:24:44.48,0:24:49.68,Default,,0000,0000,0000,,expansion. The difference of\Ntwo squares. This is very
Dialogue: 0,0:24:49.68,0:24:53.36,Default,,0000,0000,0000,,helpful sometimes when you\Nhave calculations involving
Dialogue: 0,0:24:53.36,0:24:54.40,Default,,0000,0000,0000,,sired serves.
Dialogue: 0,0:24:56.85,0:25:03.53,Default,,0000,0000,0000,,Night, sometimes you have surged\Nwritten like this one over the
Dialogue: 0,0:25:03.53,0:25:05.96,Default,,0000,0000,0000,,square root of 13.
Dialogue: 0,0:25:06.93,0:25:11.35,Default,,0000,0000,0000,,Now really, we don't like\Nanswers being given such that
Dialogue: 0,0:25:11.35,0:25:16.65,Default,,0000,0000,0000,,the denominator is assert.\NWhat we want to do is give an
Dialogue: 0,0:25:16.65,0:25:20.19,Default,,0000,0000,0000,,answer with a whole number as\Nthe denominator.
Dialogue: 0,0:25:21.25,0:25:25.68,Default,,0000,0000,0000,,Now there is a technique that\Nhelps us to change this sired
Dialogue: 0,0:25:25.68,0:25:28.63,Default,,0000,0000,0000,,into whole number and that\Ntechnique is called
Dialogue: 0,0:25:28.63,0:25:32.69,Default,,0000,0000,0000,,rationalization. What we have to\Ndo is rationalize the Route 30
Dialogue: 0,0:25:32.69,0:25:38.22,Default,,0000,0000,0000,,we have to make it into a whole\Nnumber. Now the easy way to do
Dialogue: 0,0:25:38.22,0:25:42.65,Default,,0000,0000,0000,,that is just use that simple\Nfact that keeps cropping up time
Dialogue: 0,0:25:42.65,0:25:47.08,Default,,0000,0000,0000,,and time again is that we have\Nto multiply a thigh itself.
Dialogue: 0,0:25:47.79,0:25:52.74,Default,,0000,0000,0000,,But we don't want to change the\Nvalue of this fraction. So what
Dialogue: 0,0:25:52.74,0:25:55.79,Default,,0000,0000,0000,,we have to do is multiply it by.
Dialogue: 0,0:25:56.36,0:26:00.92,Default,,0000,0000,0000,,The fraction square root of 13\Nover the square root of 13
Dialogue: 0,0:26:00.92,0:26:02.82,Default,,0000,0000,0000,,because that fraction is equal
Dialogue: 0,0:26:02.82,0:26:06.54,Default,,0000,0000,0000,,to 1. Now when we do
Dialogue: 0,0:26:06.54,0:26:08.98,Default,,0000,0000,0000,,that. And evaluated.
Dialogue: 0,0:26:09.57,0:26:14.44,Default,,0000,0000,0000,,The square root of 13 times the\Nsquare root of 13 is there Ting
Dialogue: 0,0:26:14.44,0:26:19.66,Default,,0000,0000,0000,,and on top one times the square\Nroot of 13 is the square root of
Dialogue: 0,0:26:19.66,0:26:25.08,Default,,0000,0000,0000,,30. And this is the more\Nacceptable form of writing one
Dialogue: 0,0:26:25.08,0:26:27.71,Default,,0000,0000,0000,,over the square root of 30.
Dialogue: 0,0:26:28.42,0:26:32.10,Default,,0000,0000,0000,,It's where you have, as I\Nsaid before, a whole number
Dialogue: 0,0:26:32.10,0:26:33.11,Default,,0000,0000,0000,,as the denominator.
Dialogue: 0,0:26:34.47,0:26:39.51,Default,,0000,0000,0000,,Now we can get more complicated\Nfractions, which involves
Dialogue: 0,0:26:39.51,0:26:46.23,Default,,0000,0000,0000,,serves. What if we had this\Nfraction one over 1 plus the
Dialogue: 0,0:26:46.23,0:26:48.47,Default,,0000,0000,0000,,square root of 2?
Dialogue: 0,0:26:49.54,0:26:54.19,Default,,0000,0000,0000,,Hi can be rationalized that high\Ncan we make this?
Dialogue: 0,0:26:54.76,0:26:57.31,Default,,0000,0000,0000,,Addition into a whole number.
Dialogue: 0,0:26:58.34,0:27:03.17,Default,,0000,0000,0000,,Will remember what we've just\Ndone on the previous examples.
Dialogue: 0,0:27:03.69,0:27:05.78,Default,,0000,0000,0000,,We use the difference of two
Dialogue: 0,0:27:05.78,0:27:09.88,Default,,0000,0000,0000,,squares. And the difference two\Nsquares had an expansion where
Dialogue: 0,0:27:09.88,0:27:13.36,Default,,0000,0000,0000,,you had the two numbers the\Nsame, but the operation was
Dialogue: 0,0:27:13.36,0:27:18.49,Default,,0000,0000,0000,,different. So what if we\Nmultiplied the one plus the
Dialogue: 0,0:27:18.49,0:27:22.11,Default,,0000,0000,0000,,square root of 2 by 1 minus root
Dialogue: 0,0:27:22.11,0:27:26.73,Default,,0000,0000,0000,,2? Now what we've got on the\Nbottom is the expansion of
Dialogue: 0,0:27:26.73,0:27:29.73,Default,,0000,0000,0000,,the difference of two\Nsquares, and we know that
Dialogue: 0,0:27:29.73,0:27:31.73,Default,,0000,0000,0000,,will give us a whole number.
Dialogue: 0,0:27:32.89,0:27:38.45,Default,,0000,0000,0000,,But we want to leave that this\Nfraction has the same value, so
Dialogue: 0,0:27:38.45,0:27:43.59,Default,,0000,0000,0000,,we have to multiply it by the\Nfraction 1 minus root 2.
Dialogue: 0,0:27:44.80,0:27:50.72,Default,,0000,0000,0000,,Over 1 minus root 2 because\Nthat fraction equals 1.
Dialogue: 0,0:27:52.75,0:27:57.28,Default,,0000,0000,0000,,So we evaluate what we've got\Nnow. Remember, this was the
Dialogue: 0,0:27:57.28,0:28:02.64,Default,,0000,0000,0000,,expansion of two squares and one\Nof the two squares is 1 squared
Dialogue: 0,0:28:02.64,0:28:05.11,Default,,0000,0000,0000,,and the square root of 2.
Dialogue: 0,0:28:06.20,0:28:11.37,Default,,0000,0000,0000,,1 squared minus the\Nsquare root of 2.
Dialogue: 0,0:28:12.66,0:28:16.73,Default,,0000,0000,0000,,That's what this\Nproduct is when we
Dialogue: 0,0:28:16.73,0:28:20.81,Default,,0000,0000,0000,,evaluate that that\Nequals 1 - 2.
Dialogue: 0,0:28:22.01,0:28:24.56,Default,,0000,0000,0000,,Which equals negative one.
Dialogue: 0,0:28:27.34,0:28:29.83,Default,,0000,0000,0000,,Now we simplify all of this.
Dialogue: 0,0:28:30.33,0:28:37.51,Default,,0000,0000,0000,,1 * 1 minus square root of 2\Nis 1 minus the square root of 2
Dialogue: 0,0:28:37.51,0:28:39.31,Default,,0000,0000,0000,,all over negative one.
Dialogue: 0,0:28:40.59,0:28:42.64,Default,,0000,0000,0000,,Now that doesn't look really
Dialogue: 0,0:28:42.64,0:28:48.13,Default,,0000,0000,0000,,neat. What we will do now is\Nmeeting it up. We divide through
Dialogue: 0,0:28:48.13,0:28:52.30,Default,,0000,0000,0000,,by the negative one and we\Ndivide through by the negative
Dialogue: 0,0:28:52.30,0:28:56.78,Default,,0000,0000,0000,,one. We get negative one into\None which is negative one.
Dialogue: 0,0:28:58.21,0:29:00.50,Default,,0000,0000,0000,,Than negative one into the.
Dialogue: 0,0:29:01.04,0:29:06.12,Default,,0000,0000,0000,,Negative Route 2 will give us\NPlus Route 2.
Dialogue: 0,0:29:07.10,0:29:11.42,Default,,0000,0000,0000,,And we'll just rewrite it\Nso that we've got the
Dialogue: 0,0:29:11.42,0:29:14.44,Default,,0000,0000,0000,,positive number. First,\NRoute 2 subtract 1.
Dialogue: 0,0:29:15.77,0:29:20.02,Default,,0000,0000,0000,,So when we rationalize this\Nfraction, one over 1 plus the
Dialogue: 0,0:29:20.02,0:29:25.81,Default,,0000,0000,0000,,square root of 2, you get the\Nsquare root of 2 - 1 is not
Dialogue: 0,0:29:25.81,0:29:30.05,Default,,0000,0000,0000,,really neat. It looks great.\NYou've got a fraction here, but
Dialogue: 0,0:29:30.05,0:29:31.98,Default,,0000,0000,0000,,here you've just got a
Dialogue: 0,0:29:31.98,0:29:34.62,Default,,0000,0000,0000,,subtraction. 1
Dialogue: 0,0:29:34.62,0:29:39.78,Default,,0000,0000,0000,,final example. More\Ndifficult than the previous
Dialogue: 0,0:29:39.78,0:29:44.65,Default,,0000,0000,0000,,ones, see how we can cope with\Nthis one over the square root of
Dialogue: 0,0:29:44.65,0:29:47.09,Default,,0000,0000,0000,,5 minus the square root of 3.
Dialogue: 0,0:29:47.86,0:29:49.96,Default,,0000,0000,0000,,We want to rationalize this
Dialogue: 0,0:29:49.96,0:29:53.74,Default,,0000,0000,0000,,fraction. Think of the\Ndifference of two squares.
Dialogue: 0,0:29:54.50,0:29:57.79,Default,,0000,0000,0000,,We've got ripped 5 minus Route
Dialogue: 0,0:29:57.79,0:30:05.40,Default,,0000,0000,0000,,3. If we multiply it by\NRoute 5 Plus Route 3, we know
Dialogue: 0,0:30:05.40,0:30:09.58,Default,,0000,0000,0000,,that that expansion is the\Ndifference two squares.
Dialogue: 0,0:30:10.50,0:30:14.37,Default,,0000,0000,0000,,We don't want to change the\Nvalue of the original fraction,
Dialogue: 0,0:30:14.37,0:30:18.60,Default,,0000,0000,0000,,so we have to multiply it by\Nanother fraction which equals 1
Dialogue: 0,0:30:18.60,0:30:24.23,Default,,0000,0000,0000,,and that will be square root of\N5 + 3 over the square root of 5
Dialogue: 0,0:30:24.23,0:30:26.34,Default,,0000,0000,0000,,plus the square root of 3.
Dialogue: 0,0:30:28.18,0:30:30.16,Default,,0000,0000,0000,,We work guys.
Dialogue: 0,0:30:30.78,0:30:34.60,Default,,0000,0000,0000,,This expansion using the\Ndifference of two squares, the
Dialogue: 0,0:30:34.60,0:30:38.43,Default,,0000,0000,0000,,two squares involved are the\Nsquare root of 5.
Dialogue: 0,0:30:38.96,0:30:45.91,Default,,0000,0000,0000,,Subtract the square\Nroot of 3.
Dialogue: 0,0:30:48.45,0:30:52.12,Default,,0000,0000,0000,,When we evaluate that that is 5
Dialogue: 0,0:30:52.12,0:30:56.01,Default,,0000,0000,0000,,- 3. Which equals 2.
Dialogue: 0,0:30:58.10,0:31:05.32,Default,,0000,0000,0000,,So when we rewrite all of this\None times, the square root of 5
Dialogue: 0,0:31:05.32,0:31:10.48,Default,,0000,0000,0000,,plus the square root of 3 on\Ntop over 2.
Dialogue: 0,0:31:11.53,0:31:13.12,Default,,0000,0000,0000,,We tally it up a little bit.
Dialogue: 0,0:31:13.68,0:31:17.54,Default,,0000,0000,0000,,By dividing through by the\Ntwo and so we get the square
Dialogue: 0,0:31:17.54,0:31:21.73,Default,,0000,0000,0000,,root of 5 over 2 plus the\Nsquare root of 3 over 2.
Dialogue: 0,0:31:23.68,0:31:26.17,Default,,0000,0000,0000,,So that's it. That's how you
Dialogue: 0,0:31:26.17,0:31:29.87,Default,,0000,0000,0000,,rationalize. Search You have to.
Dialogue: 0,0:31:30.37,0:31:33.69,Default,,0000,0000,0000,,Make this error denominator into\Na whole number.
Dialogue: 0,0:31:34.53,0:31:37.11,Default,,0000,0000,0000,,By multiplying the fraction.
Dialogue: 0,0:31:37.83,0:31:40.29,Default,,0000,0000,0000,,By another fraction,\Nwhich is one.
Dialogue: 0,0:31:41.84,0:31:44.89,Default,,0000,0000,0000,,And it's helpful to remember the\Ndifference of two squares.
Dialogue: 0,0:31:45.63,0:31:50.96,Default,,0000,0000,0000,,Now I want to want to do is\Nfinish off by giving you my top
Dialogue: 0,0:31:50.96,0:31:55.57,Default,,0000,0000,0000,,tents on serves. First of all\Nrecap what is assert assert is a
Dialogue: 0,0:31:55.57,0:31:57.34,Default,,0000,0000,0000,,fractional root of a whole
Dialogue: 0,0:31:57.34,0:32:02.72,Default,,0000,0000,0000,,number. Like a square root or\Ncube root, the gives you an
Dialogue: 0,0:32:02.72,0:32:06.51,Default,,0000,0000,0000,,irrational number and remember\Nan irrational number is a number
Dialogue: 0,0:32:06.51,0:32:10.68,Default,,0000,0000,0000,,that cannot be written as a\Nwhole number or a fraction.
Dialogue: 0,0:32:11.33,0:32:16.20,Default,,0000,0000,0000,,Remember there are common serves\Nthat we use a lot as route to
Dialogue: 0,0:32:16.20,0:32:18.83,Default,,0000,0000,0000,,Route 3, Route 5 and so on.
Dialogue: 0,0:32:19.85,0:32:24.79,Default,,0000,0000,0000,,And there's certain basic\Nrules that we know that are
Dialogue: 0,0:32:24.79,0:32:28.74,Default,,0000,0000,0000,,useful when we are doing\Ncalculations involving Surds.
Dialogue: 0,0:32:29.99,0:32:33.45,Default,,0000,0000,0000,,One of them is if you have the\Nsquare root of product.
Dialogue: 0,0:32:34.56,0:32:36.91,Default,,0000,0000,0000,,Then that can equal the square
Dialogue: 0,0:32:36.91,0:32:41.73,Default,,0000,0000,0000,,root. Of one number times by the\Nsquare root of the other number.
Dialogue: 0,0:32:42.77,0:32:45.79,Default,,0000,0000,0000,,And similarly, the square root\Nof a quotient.
Dialogue: 0,0:32:46.93,0:32:51.39,Default,,0000,0000,0000,,Is the square root of 1 number\Ndivided by the other square
Dialogue: 0,0:32:51.39,0:32:56.26,Default,,0000,0000,0000,,root? But remember the\Nsquare root of addition is
Dialogue: 0,0:32:56.26,0:33:01.05,Default,,0000,0000,0000,,not the sum of the square\Nroots that a common mistake,
Dialogue: 0,0:33:01.05,0:33:02.80,Default,,0000,0000,0000,,so don't make it.
Dialogue: 0,0:33:04.28,0:33:07.78,Default,,0000,0000,0000,,Another technique that we use
Dialogue: 0,0:33:07.78,0:33:10.70,Default,,0000,0000,0000,,insert calculations. Is
Dialogue: 0,0:33:10.70,0:33:17.16,Default,,0000,0000,0000,,rationalization? We don't like\Nsurge being the denominators of
Dialogue: 0,0:33:17.16,0:33:21.46,Default,,0000,0000,0000,,fractions. So what we have to do\Nis rationalize the fraction.
Dialogue: 0,0:33:22.28,0:33:26.18,Default,,0000,0000,0000,,We multiply by a fraction that\Nis equal to 1.
Dialogue: 0,0:33:27.34,0:33:29.80,Default,,0000,0000,0000,,And sometimes when\Nwe have to do that.
Dialogue: 0,0:33:30.91,0:33:35.51,Default,,0000,0000,0000,,The difference of two squares is\Nuseful. The expansion of the
Dialogue: 0,0:33:35.51,0:33:37.60,Default,,0000,0000,0000,,difference of two squares is.
Dialogue: 0,0:33:38.17,0:33:44.42,Default,,0000,0000,0000,,A squared minus B squared is a\Nminus B Times A+B.
Dialogue: 0,0:33:45.40,0:33:50.20,Default,,0000,0000,0000,,Now, if you know all of those\Ntop tips, I think you'd be
Dialogue: 0,0:33:50.20,0:33:52.78,Default,,0000,0000,0000,,pretty good in doing\Ncalculations involving surds.