[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.61,0:00:03.30,Default,,0000,0000,0000,,We're asked to subtract all of this\Ncraziness
Dialogue: 0,0:00:03.30,0:00:05.63,Default,,0000,0000,0000,,over here, and it looks daunting, but if
Dialogue: 0,0:00:05.63,0:00:08.46,Default,,0000,0000,0000,,we really just focus, it actually should\Nbe
Dialogue: 0,0:00:08.46,0:00:11.06,Default,,0000,0000,0000,,pretty straightforward to subtract and\Nsimplify this thing.
Dialogue: 0,0:00:11.06,0:00:17.08,Default,,0000,0000,0000,,Cuz right from the get go, I have 4 times\Nthe fourth root of 81x to the fifth,
Dialogue: 0,0:00:17.08,0:00:23.18,Default,,0000,0000,0000,,and from that I wanna subtract 2 times the\Nfourth root of 81x to the fifth.
Dialogue: 0,0:00:23.18,0:00:26.60,Default,,0000,0000,0000,,And so, you really can just say, lookI\Nhave four of something and this
Dialogue: 0,0:00:26.60,0:00:30.46,Default,,0000,0000,0000,,something, I'll just circle in yellow I\Nhave four of this, it could be lemons.
Dialogue: 0,0:00:30.46,0:00:33.80,Default,,0000,0000,0000,,I have four of these things and I wanna\Nsubtract two of these things.
Dialogue: 0,0:00:33.80,0:00:35.84,Default,,0000,0000,0000,,These are the exact same things.
Dialogue: 0,0:00:35.84,0:00:39.65,Default,,0000,0000,0000,,They're the 4th root of 81x to the 5th,\N4th root of 81x to the 5th.
Dialogue: 0,0:00:39.65,0:00:42.53,Default,,0000,0000,0000,,So if I have four of, if I have four\Nlemons and
Dialogue: 0,0:00:42.53,0:00:46.17,Default,,0000,0000,0000,,I wanna subtract two lemons, I'm gonna\Nhave two lemons left over.
Dialogue: 0,0:00:46.17,0:00:47.81,Default,,0000,0000,0000,,Or if I have four of this thing and I take\Naway
Dialogue: 0,0:00:47.81,0:00:51.74,Default,,0000,0000,0000,,two of this thing, I'm gonna have two of\Nthese things left over.
Dialogue: 0,0:00:51.74,0:00:56.03,Default,,0000,0000,0000,,So these terms right over here simplify to\N2
Dialogue: 0,0:00:56.03,0:01:00.45,Default,,0000,0000,0000,,times the fouthth root of 81x to the\Nfifth.
Dialogue: 0,0:01:00.45,0:01:03.31,Default,,0000,0000,0000,,And I got this 2 just by subtracting the\Ncoefficients 4
Dialogue: 0,0:01:03.31,0:01:07.30,Default,,0000,0000,0000,,something minus 2 something is equal to 2\Nof that something.
Dialogue: 0,0:01:07.30,0:01:11.56,Default,,0000,0000,0000,,And then that, of course, we still have\Nthis minus the regular
Dialogue: 0,0:01:11.56,0:01:16.16,Default,,0000,0000,0000,,principal square root of x to the third,\Nof x to the third.
Dialogue: 0,0:01:16.16,0:01:18.45,Default,,0000,0000,0000,,Now I wanted to try to simplify, I wanna\Ntry to
Dialogue: 0,0:01:18.45,0:01:21.66,Default,,0000,0000,0000,,simplify what's inside of these under the\Nradical signs, so that
Dialogue: 0,0:01:21.66,0:01:24.19,Default,,0000,0000,0000,,we can, on this, in this example actually\Ntake the fourth
Dialogue: 0,0:01:24.19,0:01:29.24,Default,,0000,0000,0000,,root and over here actually take, maybe, a\Nprincipal square root.
Dialogue: 0,0:01:29.24,0:01:33.61,Default,,0000,0000,0000,,So first of all, let's see if 81 either is\Na, is something to the fourth power,
Dialogue: 0,0:01:33.61,0:01:35.21,Default,,0000,0000,0000,,or at least can be factored into something
Dialogue: 0,0:01:35.21,0:01:37.77,Default,,0000,0000,0000,,that is a, a something to the fourth\Npower.
Dialogue: 0,0:01:37.77,0:01:42.61,Default,,0000,0000,0000,,So 81, if we do prime factorization, is 3\Ntimes 27,
Dialogue: 0,0:01:42.61,0:01:45.51,Default,,0000,0000,0000,,27 is 3 times 9, and 9 is 3 times 3.
Dialogue: 0,0:01:45.51,0:01:49.61,Default,,0000,0000,0000,,So 81 is exactly 3 times 3 times 3 times\N3.
Dialogue: 0,0:01:49.61,0:01:53.01,Default,,0000,0000,0000,,So 81 actually is 3 to the fourth power\Nwhich is
Dialogue: 0,0:01:53.01,0:01:56.31,Default,,0000,0000,0000,,convenient, cuz we're gonna be taking the\Nfourth root of that.
Dialogue: 0,0:01:56.31,0:01:59.58,Default,,0000,0000,0000,,And then x to the fifth we can write as a\Nproduct.
Dialogue: 0,0:01:59.58,0:02:01.57,Default,,0000,0000,0000,,We can, let me write it over here so it\Ndoesn't get messy.
Dialogue: 0,0:02:01.57,0:02:05.75,Default,,0000,0000,0000,,So I'm gonna write what's under the\Nradical as 3 to
Dialogue: 0,0:02:05.75,0:02:11.29,Default,,0000,0000,0000,,the fourth power, times, times x to the\Nfourth power times x.
Dialogue: 0,0:02:11.29,0:02:13.71,Default,,0000,0000,0000,,x to the fourth times x is x to the fifth\Npower.
Dialogue: 0,0:02:13.71,0:02:16.27,Default,,0000,0000,0000,,And, I'm taking the fourth root of all of\Nthis.
Dialogue: 0,0:02:16.27,0:02:18.33,Default,,0000,0000,0000,,And, taking the fourth root of all of\Nthis,
Dialogue: 0,0:02:18.33,0:02:20.36,Default,,0000,0000,0000,,that's the same thing as taking the fourth\Nroot
Dialogue: 0,0:02:20.36,0:02:23.72,Default,,0000,0000,0000,,of this and, taking the fourth root of\Nthis,
Dialogue: 0,0:02:23.72,0:02:25.33,Default,,0000,0000,0000,,and let me just, I'm gonna just skip step.
Dialogue: 0,0:02:25.33,0:02:27.50,Default,,0000,0000,0000,,So, I'm taking the fourth root.
Dialogue: 0,0:02:27.50,0:02:30.73,Default,,0000,0000,0000,,I'm taking the fourth root of all of it,\Nright over there.
Dialogue: 0,0:02:30.73,0:02:32.93,Default,,0000,0000,0000,,And of course I have a 2 out front.
Dialogue: 0,0:02:32.93,0:02:36.38,Default,,0000,0000,0000,,And then x to the third can be written as\Nx squared times x.
Dialogue: 0,0:02:36.38,0:02:41.54,Default,,0000,0000,0000,,It's minus the principle square root of x\Nsquared times x.
Dialogue: 0,0:02:41.54,0:02:45.78,Default,,0000,0000,0000,,And I broke it up like this cuz this,\Nright over here, is a perfect square.
Dialogue: 0,0:02:45.78,0:02:48.57,Default,,0000,0000,0000,,Now, how can we simplify this a little\Nbit?
Dialogue: 0,0:02:48.57,0:02:50.48,Default,,0000,0000,0000,,And you're probably getting used to the\Npattern.
Dialogue: 0,0:02:50.48,0:02:54.22,Default,,0000,0000,0000,,This is the same thing as the fourth root\Nafter you get the fourth,
Dialogue: 0,0:02:54.22,0:02:58.41,Default,,0000,0000,0000,,times the fourth root of x to the fourth,\Ntimes the fourth root of x.
Dialogue: 0,0:02:58.41,0:02:59.97,Default,,0000,0000,0000,,So let's just skip straight to that.
Dialogue: 0,0:02:59.97,0:03:02.80,Default,,0000,0000,0000,,So what is, what is the fourth root?
Dialogue: 0,0:03:02.80,0:03:04.49,Default,,0000,0000,0000,,Well I ca, I can write it, let me write
Dialogue: 0,0:03:04.49,0:03:06.38,Default,,0000,0000,0000,,it explicitly, although you wouldn't have\Nto necessarily do this.
Dialogue: 0,0:03:06.38,0:03:12.74,Default,,0000,0000,0000,,This is the same thing as the fourth, as\Nthe fourth root of 3 to the fourth,
Dialogue: 0,0:03:12.74,0:03:18.27,Default,,0000,0000,0000,,times the fourth root of x to the fourth,\Ntimes the fourth root
Dialogue: 0,0:03:18.27,0:03:24.48,Default,,0000,0000,0000,,of x, times the fourth root of x, and 2 as\Nbeing multiplied times all of that.
Dialogue: 0,0:03:24.48,0:03:27.55,Default,,0000,0000,0000,,And then this over here is minus the\Nprinciple square
Dialogue: 0,0:03:27.55,0:03:32.30,Default,,0000,0000,0000,,root of x squared, times the principle\Nsquare root of x.
Dialogue: 0,0:03:32.30,0:03:34.11,Default,,0000,0000,0000,,And so, if we try to simplify it, the\Nfourth
Dialogue: 0,0:03:34.11,0:03:37.55,Default,,0000,0000,0000,,root of 3 to the fourth power is just 3.
Dialogue: 0,0:03:37.55,0:03:43.83,Default,,0000,0000,0000,,So, we get a 3 there, the fourth root of x\Nto the fourth power is just going to be
Dialogue: 0,0:03:43.83,0:03:49.61,Default,,0000,0000,0000,,x, is just going to be is just, actually,\Nlook,
Dialogue: 0,0:03:49.61,0:03:51.15,Default,,0000,0000,0000,,I just reminded myself, you have to be\Ncareful there.
Dialogue: 0,0:03:51.15,0:03:54.41,Default,,0000,0000,0000,,It is not just x, because what if x is\Nnegative?
Dialogue: 0,0:03:54.41,0:03:56.72,Default,,0000,0000,0000,,If x is negative, then x to the fourth\Npower is going
Dialogue: 0,0:03:56.72,0:03:59.60,Default,,0000,0000,0000,,to be a positive value, and when you take\Nthe fourth, remember, this
Dialogue: 0,0:03:59.60,0:04:02.83,Default,,0000,0000,0000,,is the fourth principle root, you're going\Nto get the positive version
Dialogue: 0,0:04:02.83,0:04:06.26,Default,,0000,0000,0000,,of x, or really, you're going to get the\Nabsolute value of x.
Dialogue: 0,0:04:06.26,0:04:08.45,Default,,0000,0000,0000,,So here you're going to be getting, you're
Dialogue: 0,0:04:08.45,0:04:12.59,Default,,0000,0000,0000,,going to be getting the absolute value of\Nx.
Dialogue: 0,0:04:12.59,0:04:15.62,Default,,0000,0000,0000,,And then, although, well you could make an\Nargument
Dialogue: 0,0:04:15.62,0:04:18.83,Default,,0000,0000,0000,,that x needs to be positive if this thing
Dialogue: 0,0:04:18.83,0:04:21.25,Default,,0000,0000,0000,,is going to be well-defined in the real\Nnumbers,
Dialogue: 0,0:04:21.25,0:04:23.13,Default,,0000,0000,0000,,cuz then what's under the radical has to\Nbe positive.
Dialogue: 0,0:04:23.13,0:04:25.77,Default,,0000,0000,0000,,But let's just go with this for right now.
Dialogue: 0,0:04:25.77,0:04:28.57,Default,,0000,0000,0000,,And then we have the fourth root of x.
Dialogue: 0,0:04:28.57,0:04:30.87,Default,,0000,0000,0000,,And then over here the, the principal\Nsquare of
Dialogue: 0,0:04:30.87,0:04:34.55,Default,,0000,0000,0000,,x squared by the same logic, by the same\Nlogic
Dialogue: 0,0:04:34.55,0:04:36.44,Default,,0000,0000,0000,,is going to be the absolute value of x and
Dialogue: 0,0:04:36.44,0:04:38.71,Default,,0000,0000,0000,,then this is just the principal square\Nroot of x.
Dialogue: 0,0:04:38.71,0:04:40.03,Default,,0000,0000,0000,,So, let's multiply everything out.
Dialogue: 0,0:04:40.03,0:04:44.48,Default,,0000,0000,0000,,We have 2 times 3 times the absolute value\Nof x.
Dialogue: 0,0:04:44.48,0:04:50.64,Default,,0000,0000,0000,,So, 2 times 3 is 6 times the absolute\Nvalue of x, times the principal
Dialogue: 0,0:04:50.64,0:04:56.49,Default,,0000,0000,0000,,or the, the principal fourth root of x, I\Nshould say, minus,
Dialogue: 0,0:04:56.49,0:05:02.47,Default,,0000,0000,0000,,minus, we've took out the absolute value\Nof x, times the principle root of x.
Dialogue: 0,0:05:02.47,0:05:04.39,Default,,0000,0000,0000,,And we can't do anymore subtracting.
Dialogue: 0,0:05:04.39,0:05:06.75,Default,,0000,0000,0000,,Just because you have to realize, this is\Na fourth
Dialogue: 0,0:05:06.75,0:05:10.71,Default,,0000,0000,0000,,root, this is a regular square root,\Nprinciple square root.
Dialogue: 0,0:05:10.71,0:05:14.87,Default,,0000,0000,0000,,If these were the same root, then maybe we\Ncould simplify this a little bit more.
Dialogue: 0,0:05:14.87,0:05:18.20,Default,,0000,0000,0000,,And so then we are all done, and we have\Nfully simplified it.
Dialogue: 0,0:05:18.20,0:05:21.13,Default,,0000,0000,0000,,And if you make the assumption that this\Nis defined
Dialogue: 0,0:05:21.13,0:05:24.25,Default,,0000,0000,0000,,for real numbers so that the domain over\Nhere this would
Dialogue: 0,0:05:24.25,0:05:27.17,Default,,0000,0000,0000,,has to be under these radicals has to be\Npositive
Dialogue: 0,0:05:27.17,0:05:29.86,Default,,0000,0000,0000,,actually everyone of these cases and if\Nthere need to be
Dialogue: 0,0:05:29.86,0:05:32.35,Default,,0000,0000,0000,,positive not gonna be dealing with\Nimaginary numbers all of
Dialogue: 0,0:05:32.35,0:05:35.05,Default,,0000,0000,0000,,these need to be positive, their domains\Nare x has to
Dialogue: 0,0:05:35.05,0:05:36.98,Default,,0000,0000,0000,,be greater than or equal to 0 then you\Ncould assume
Dialogue: 0,0:05:36.98,0:05:39.43,Default,,0000,0000,0000,,that the absolute value of x is the same\Nas x.
Dialogue: 0,0:05:39.43,0:05:41.39,Default,,0000,0000,0000,,But I will just take it right here, if you\Nrestrict
Dialogue: 0,0:05:41.39,0:05:43.95,Default,,0000,0000,0000,,the domain you could get rid of the\Nabsolute value sign.