WEBVTT 00:00:05.000 --> 00:00:07.680 Neither of these two terms simplify. 00:00:09.080 --> 00:00:11.420 Are we able to combine these together? 00:00:14.680 --> 00:00:17.040 Yeah, are we able to add them? 00:00:17.440 --> 00:00:18.340 What do they add to? 00:00:21.840 --> 00:00:23.400 Six square root three. 00:00:24.000 --> 00:00:24.700 Why is that? 00:00:24.700 --> 00:00:26.800 They both have a factor of square root three. 00:00:28.480 --> 00:00:30.320 So we could factor that out. 00:00:31.300 --> 00:00:35.860 You get 2 plus 4 and that is 6 square root 3. 00:00:38.840 --> 00:00:45.579 The radicals, the way we add them is you add the coefficients and then multiply by 00:00:45.579 --> 00:00:46.000 what's 00:00:46.000 --> 00:00:46.900 You could factor out the x. 00:00:47.900 --> 00:00:49.580 Ok, these are called like radicals. 00:00:52.080 --> 00:00:53.480 Oops. 00:01:00.540 --> 00:01:03.360 Three square root two. Which other term is a like radical? 00:01:05.640 --> 00:01:11.920 Think of this as plus negative six square root two. 00:01:13.860 --> 00:01:20.492 Three square root two plus negative six square root two is negative three square 00:01:20.492 --> 00:01:21.440 root two. 00:01:22.300 --> 00:01:29.340 Four square root five plus seven square root five is eleven square root five. 00:01:29.340 --> 00:01:38.020 you're done the directions are either simplify or it might just say add okay 00:01:38.020 --> 00:01:46.680 These are not like radicals yet, but both radicals simplify. 00:01:46.680 --> 00:01:54.280 these simplify I think so so on the side we start simplifying eight is two four 00:01:54.280 --> 00:01:59.300 two two this is two square root two 00:02:07.760 --> 00:02:10.760 And this is 5 square root 2 00:02:11.620 --> 00:02:16.201 Now once we simplify and we see they happen to be like radicals so we can now 00:02:16.201 --> 00:02:16.951 combine them 00:02:17.820 --> 00:02:22.800 Two square root two plus five square root two is seven square root two. 00:02:23.480 --> 00:02:25.820 2 plus 5 7 2 00:02:35.560 --> 00:02:40.560 So first we simplify all the radicals and then we combine all the like radicals. 00:02:41.060 --> 00:02:49.560 Makes sense? Okay. And subtracting, like I said, same thing as adding a negative so 00:02:49.560 --> 00:02:51.280 It's the same exact thing. 00:02:51.820 --> 00:02:53.100 So now we're multiplying. 00:02:57.540 --> 00:03:01.688 If there's more than one term, terms or things separated by addition, what property 00:03:01.688 --> 00:03:02.280 do we 00:03:02.280 --> 00:03:03.180 have to use? 00:03:06.980 --> 00:03:07.880 What did you just say? 00:03:08.360 --> 00:03:09.100 Sorry. 00:03:09.100 --> 00:03:14.080 So here we're multiplying, but this has more than one term. 00:03:15.080 --> 00:03:16.680 Terms or things separated by addition. 00:03:18.340 --> 00:03:23.945 What property do we have to use? Distributive. What's the long name? Did we talk 00:03:23.945 --> 00:03:25.440 about that last week? 00:03:30.680 --> 00:03:34.740 Distributive property of multiplication over addition. 00:03:34.740 --> 00:03:37.480 So we distribute. 00:03:50.740 --> 00:03:55.109 And then that's two rad, square root three times square root three is square root 00:03:55.109 --> 00:03:55.859 nine. 00:03:57.580 --> 00:03:59.700 Square root three times square root six is square root eighteen. 00:04:01.280 --> 00:04:03.920 and then we just start simplifying 00:04:07.620 --> 00:04:08.860 18 is 00:04:17.280 --> 00:04:20.140 Square root eighteen simplifies to three square root two. 00:04:23.500 --> 00:04:27.720 The result is six plus three square root two. 00:04:27.720 --> 00:04:29.720 We're at two plus six. 00:04:37.720 --> 00:04:43.940 You could first write two square root one hundred fifty, then simplify. 00:04:43.940 --> 00:04:51.995 at 150, but I noticed right, 5 was a factor, so I wrote it as just 5 times 5 times 00:04:51.995 --> 00:04:52.800 6, and 00:04:52.800 --> 00:04:59.063 then that's a perfect square square root 25 is 5 just kind of saves you time a 00:04:59.063 --> 00:04:59.813 little bit 00:05:02.840 --> 00:05:19.913 okay questions on that question did I screw up sorry six times to square root 10 00:05:19.913 --> 00:05:20.980 let's 00:05:20.980 --> 00:05:23.300 go 12 there we go thank you 00:05:23.300 --> 00:05:28.340 Now we have two terms times two terms. 00:05:30.360 --> 00:05:32.540 What property are we going to have to use here? 00:05:36.420 --> 00:05:40.620 We are still using the distributive property of multiplication. 00:05:42.140 --> 00:05:50.067 But instead of distributing one term to each thing separated by the addition, we 00:05:50.067 --> 00:05:51.200 have to 00:05:51.200 --> 00:05:59.340 distribute the entire thing to each term. 00:06:00.780 --> 00:06:09.980 Distribute the entire first binomial to each term of the second binomial. 00:06:19.580 --> 00:06:24.920 So you distribute both terms, or the... 00:06:24.920 --> 00:06:31.941 And now once you distribute both terms each term here, okay now we have to use the 00:06:31.941 --> 00:06:33.180 distributive property again 00:06:35.420 --> 00:06:37.940 Right here teach turn 00:06:43.480 --> 00:06:46.480 So we got two times around five 00:06:57.760 --> 00:07:03.980 Two terms times two terms produce four terms before combining. 00:07:03.980 --> 00:07:08.700 and then we combine any like terms or like radicals. 00:07:08.700 --> 00:07:11.760 Is that wrong? 00:07:24.920 --> 00:07:26.660 And are any of these like radicals? 00:07:27.860 --> 00:07:28.900 No. So we're done. 00:07:37.120 --> 00:07:40.780 Where did this subtraction come from? That sign needs to be corrected. 00:07:41.940 --> 00:07:45.500 We have to distribute to both terms, keeping the negative with the four. 00:07:51.920 --> 00:07:57.780 So yes, this right here, we have to distribute to both of these. 00:07:57.780 --> 00:08:05.940 thank you for picking up on that that was bad I usually just leave the 00:08:05.940 --> 00:08:08.720 negative with the four and then just do plus then we definitely would not have 00:08:08.720 --> 00:08:10.120 Here we are multiplying. 00:08:11.620 --> 00:08:20.340 Again, we're going to distribute this whole thing to each term. 00:08:34.360 --> 00:08:38.540 And now we have to use the distribute property twice. 00:09:00.460 --> 00:09:01.860 Oh 00:09:30.860 --> 00:09:34.120 And this guy simplifies right. 00:09:49.520 --> 00:09:53.260 And none of these are like radicals, so we're all done. 00:09:57.080 --> 00:10:01.340 Again this is all addition, so the order of any of those terms does not matter. 00:10:01.340 --> 00:10:03.760 Oh, same thing. 00:10:03.760 --> 00:10:08.280 I'll just review this first one. 00:11:09.320 --> 00:11:11.040 The middle terms add to zero, leaving two minus sixteen, or negative fourteen. 00:11:11.040 --> 00:11:22.860 zero zero get negative 14 00:11:42.860 --> 00:11:45.760 Let's try another one. 00:11:56.300 --> 00:11:58.440 Try this one on your own. 00:13:55.300 --> 00:13:57.500 The middle terms add to zero, leaving nine minus twenty-eight, or negative nineteen. 00:13:57.500 --> 00:14:11.640 I get 9, 7, almost 28, middle 19. 00:14:29.880 --> 00:14:34.500 Both conjugate products simplify to integers. 00:14:37.860 --> 00:14:39.140 Why is that? 00:14:43.780 --> 00:14:45.760 It's like a difference of squares. 00:14:48.500 --> 00:14:50.680 Yeah, we're going to talk about that in a minute. 00:14:50.680 --> 00:14:58.920 The only difference between these two sets of two terms is one has a minus one has a 00:14:58.920 --> 00:14:59.820 plus. 00:15:02.760 --> 00:15:04.100 Same thing with this one. 00:15:05.460 --> 00:15:08.300 Rad 2 plus 4, square root 2 minus 4. 00:15:10.340 --> 00:15:13.540 Whenever you have that they are called conjugates. 00:15:14.380 --> 00:15:22.756 When you multiply two conjugates, you always end up with the middle two terms end up 00:15:22.756 --> 00:15:23.280 adding 00:15:23.280 --> 00:15:24.180 to be zero. 00:15:25.800 --> 00:15:28.840 Okay, and we're going to need that later on. 00:15:28.840 --> 00:15:35.380 uh you know define a conjugate for you later 00:15:35.380 --> 00:15:39.320 That covered adding, subtracting, and multiplying. Now we are dividing radicals. 00:15:42.480 --> 00:15:43.640 How are we going to divide? 00:16:09.600 --> 00:16:10.840 Here's what we're going to do. 00:16:10.980 --> 00:16:19.840 This process is called rationalizing the denominator. 00:16:26.100 --> 00:16:27.820 What's the definition of a rational number? 00:16:29.100 --> 00:16:41.160 A rational number can be written as a ratio of two integers. 00:16:41.160 --> 00:16:44.240 a perfect square is an irrational number. 00:16:45.260 --> 00:16:46.480 It cannot be written as a fraction. 00:16:47.420 --> 00:16:52.440 So what we're going to do is we're going to rationalize the denominator. 00:16:52.440 --> 00:16:55.260 to turn the denominator into a rational number. 00:16:57.900 --> 00:16:59.540 And here's how we do it. 00:17:03.940 --> 00:17:05.800 We want to get rid of this square root. 00:17:06.860 --> 00:17:11.060 Multiply by square root two over square root two, which equals one. 00:17:12.700 --> 00:17:14.400 We can't just multiply by square root 2. 00:17:15.320 --> 00:17:17.820 But you can multiply by square root 2 over square root 2. 00:17:18.040 --> 00:17:18.940 Why can we do that? 00:17:18.940 --> 00:17:21.760 It's just number one. 00:17:23.120 --> 00:17:24.020 Okay. 00:17:24.980 --> 00:17:29.300 The denominator becomes four and the numerator becomes square root twelve. 00:17:31.020 --> 00:17:34.000 over two times two. 00:17:38.580 --> 00:17:41.520 So it's square root 12 over four. 00:17:44.540 --> 00:17:46.920 And then we simplify any radicals. 00:17:53.640 --> 00:17:54.698 Square root twelve is two square root three, so the fraction simplifies to square 00:17:54.698 --> 00:17:55.448 root three over two. 00:18:01.080 --> 00:18:06.200 And then the last thing we do is simplify. 00:18:07.520 --> 00:18:10.400 2 goes into the 1s. 00:18:10.900 --> 00:18:12.060 Rad 3 over 2. 00:18:15.300 --> 00:18:17.580 So we now rationalize the denominator. 00:18:17.580 --> 00:18:19.640 The denominator is a rational number. 00:18:22.160 --> 00:18:25.020 Okay, and this is basically how we're going to divide. 00:18:26.200 --> 00:18:30.040 All we do is rationalize the denominator 00:18:30.040 --> 00:18:33.180 And again, the direction might say divide or might just say simplified. 00:18:39.140 --> 00:18:41.260 And so what am I going to multiply up here? 00:18:41.260 --> 00:18:46.160 Rad 3 over square root 3. 00:18:48.140 --> 00:18:53.320 Multiply by square root three over square root three. 00:18:55.500 --> 00:18:56.980 So square root 45. 00:19:01.140 --> 00:19:02.400 And what is square root 45? 00:19:13.260 --> 00:19:15.320 Square root forty-five is three square root five, so the result is square root five. 00:19:16.500 --> 00:19:17.520 And this is where I left. 00:19:17.520 --> 00:19:19.400 Multiply by square root five over square root five. 00:19:19.400 --> 00:19:21.840 I'm going to do it. 00:19:22.180 --> 00:19:25.500 Red five by the square root five. 00:20:02.720 --> 00:20:09.020 The expression simplifies to two square root two over three. 00:20:19.580 --> 00:20:21.060 We've got questions on that. 00:20:22.220 --> 00:20:23.820 Again, we rationalized the denominator. 00:20:23.980 --> 00:20:25.280 This is not a rational number. 00:20:25.960 --> 00:20:28.200 So no, we don't want any square roots in the denominator. 00:20:30.440 --> 00:20:31.880 Let's try another one. 00:20:31.880 --> 00:20:33.340 Do not multiply this two-term denominator by square root five alone. 00:20:34.120 --> 00:20:35.780 We're not going to multiply by square root 5. 00:20:37.760 --> 00:20:44.860 Because if you do, we get 3 square root 5 plus 5. 00:20:45.760 --> 00:20:47.380 We did get rid of this square root. 00:20:49.080 --> 00:20:51.080 But now we got another one stuck here. 00:20:54.760 --> 00:20:56.120 Multiply by square root 5 again. 00:20:56.900 --> 00:20:58.600 Yeah, you should try that. 00:20:58.740 --> 00:20:59.640 Try that. 00:21:02.660 --> 00:21:10.060 Now we're at 2 times 5 is 10 over 3 times 5 plus 5 right 5. 00:21:10.620 --> 00:21:11.500 Wait, one more time. 00:21:11.500 --> 00:21:12.400 One more time? 00:21:12.500 --> 00:21:13.400 Yeah. 00:21:15.260 --> 00:21:16.300 Okay, so that doesn't work. 00:21:24.760 --> 00:21:26.760 What did somebody say over there? 00:21:28.160 --> 00:21:32.900 Multiply by the conjugate of the denominator. 00:21:36.100 --> 00:21:43.980 The conjugate of three plus square root five is three minus square root five. 00:21:43.980 --> 00:21:50.500 So we're going to multiply by 3 minus square root 5 or 3 minus square root 5. 00:21:58.280 --> 00:22:00.480 Now we've got a lot of distributing. 00:22:06.400 --> 00:22:09.440 6 minus 2 square root 5. 00:23:00.460 --> 00:23:06.600 And once you multiply by the conjugate, the two middle terms always has to be zero. 00:23:11.940 --> 00:23:18.980 After multiplying by the conjugate, the two middle denominator terms cancel. 00:23:22.380 --> 00:23:23.960 And we're almost done. 00:23:25.160 --> 00:23:26.320 Why are we not quite done? 00:23:27.740 --> 00:23:30.160 We definitely rationalized the denominator. 00:23:31.680 --> 00:23:35.340 But they all have a factor of 2 in it. 00:23:38.100 --> 00:23:39.560 Okay, you could do this. 00:23:39.740 --> 00:23:43.140 6 over 4 minus 2 square root 5 over 4. 00:23:44.420 --> 00:23:46.460 Which is 3 over 2 minus 00:23:48.300 --> 00:23:50.300 square root 5 over 2. 00:23:52.900 --> 00:23:57.540 Factor two from the numerator and simplify to three minus square root five over two. 00:24:00.700 --> 00:24:07.280 The other option is you just factor out a 2. 00:24:07.280 --> 00:24:16.500 We get 2 times 3 minus square root phi over 4. 00:24:19.400 --> 00:24:20.920 And that simplifies the one. 00:24:20.920 --> 00:24:31.380 okay questions on that so if we're dividing by something with two terms as 00:24:31.380 --> 00:24:36.499 As a radical we have to multiply by the conjugate over the conjugate of the 00:24:36.499 --> 00:24:37.249 denominator 00:24:38.820 --> 00:24:40.880 Let's try this one 00:25:12.880 --> 00:25:16.460 Are we going to multiply by the conjugate of the numerator or denominator? 00:25:17.600 --> 00:25:18.160 Denominator. 00:25:18.160 --> 00:25:20.320 Our whole goal is to get rid of square roots in the denominator. 00:25:21.660 --> 00:25:24.120 So we're going to multiply by 3. 00:25:24.460 --> 00:25:25.540 You just switch the sign. 00:25:25.540 --> 00:25:37.120 plus square root 3 plus square root 7. And now we have lots of distributing to do. 00:26:48.800 --> 00:26:50.280 Distribute carefully through the numerator and denominator. 00:26:50.300 --> 00:26:51.800 Don't mess up. 00:26:55.540 --> 00:27:03.340 plus four and six, plus seven, plus 14. 00:27:10.420 --> 00:27:12.880 It becomes three. 00:27:48.260 --> 00:27:51.140 Are any of the ones in the numerator like radicals? 00:27:55.140 --> 00:28:09.620 no that's gonna be it right there it's the best we could do okay what 00:28:09.620 --> 00:28:13.520 Adding and subtracting use like radicals; multiplying uses distribution; dividing 00:28:13.520 --> 00:28:15.080 requires rationalizing the denominator. 00:28:15.080 --> 00:28:19.900 dividing divided is the only weird one we don't really divide we multiply by 00:28:19.900 --> 00:28:27.120 something to rationalize the denominator when there's two terms you multiply by 00:28:27.120 --> 00:28:37.400 the conjugate so a plus B the conjugate 00:28:40.080 --> 00:28:47.580 is A minus B and vice versa. These are conjugates. All we do is switch the term.