WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:02.200 Welcome to Educator.com. 00:00:02.200 --> 00:00:06.000 Today, we will be covering operations with radical expressions. 00:00:06.000 --> 00:00:11.800 And in earlier lessons in Algebra I, we talked about square roots and properties when working with square roots. 00:00:11.800 --> 00:00:17.000 And now, we are going to go on to talk about other roots. 00:00:17.000 --> 00:00:20.800 I will be starting out with reviewing properties of radicals and applying these properties 00:00:20.800 --> 00:00:25.500 to roots other than square roots: cube roots, fourth roots, sixth roots, and on. 00:00:25.500 --> 00:00:29.600 It is the same properties, such as the quotient property. 00:00:29.600 --> 00:00:34.100 And the quotient property says that, if you have the root (it could be a square root; 00:00:34.100 --> 00:00:45.300 it could be a cube root) of a/b, this is equivalent to the root of a divided by the root of b, 00:00:45.300 --> 00:00:49.900 with the restriction that b cannot be equal to 0, because if b did equal 0, 00:00:49.900 --> 00:00:53.600 the square root of that would be 0 (or cube root, or whichever root). 00:00:53.600 --> 00:01:00.100 And then, you would end up with 0 in the denominator, which would result in an undefined expression. 00:01:00.100 --> 00:01:20.900 OK, so first looking at this quotient property with an example: the fifth root of 17x^10, all of that divided by 12y⁴. 00:01:20.900 --> 00:01:27.100 This is the fifth root of 17x^10, divided by 12y⁴. 00:01:27.100 --> 00:01:38.700 This can be split up into this: the fifth root of 17x^10 over the fifth root of 12y⁴. 00:01:38.700 --> 00:01:45.500 And the quotient property, along with the product property, allows us to simplify radical expressions, 00:01:45.500 --> 00:01:48.000 which we are going to talk about in a few minutes. 00:01:48.000 --> 00:01:53.700 Here, I have the product property: this would be something along the lines of 00:01:53.700 --> 00:02:10.800 the fourth root of 4x⁶ equals the fourth root of 4, times the fourth root of x⁶. 00:02:10.800 --> 00:02:18.700 And again, this helps us to simplify--the same idea as working with square roots, only now the index is a different number. 00:02:18.700 --> 00:02:23.300 There is also the restriction that, if n is even, a and b are greater than or equal to 0. 00:02:23.300 --> 00:02:30.200 And that would be to avoid this situation: look at something like the square root of 10. 00:02:30.200 --> 00:02:40.300 Using the product property, I could say, "OK, this equals the square root of 5 times 2, which equals the square root of 5, times the square root of 2." 00:02:40.300 --> 00:02:44.600 So, I have an even index, and I end up with this--not a problem. 00:02:44.600 --> 00:02:55.300 However, I could also say, "Well, the square root of 10...10 also factors out to -5 times -2." 00:02:55.300 --> 00:03:02.700 So, I follow the product property, and I end up with the square root of -5, times the square root of -2. 00:03:02.700 --> 00:03:08.500 And this is not what we are looking for when we talk about a radical; we are staying with real numbers. 00:03:08.500 --> 00:03:24.900 And so, when we have an even index (it could be 2, 4, 6, 8, something higher), then we define a and b as greater than or equal to 0, not as negative numbers. 00:03:24.900 --> 00:03:30.400 Again, reviewing some concepts that were learned in Algebra I and applied to square roots, 00:03:30.400 --> 00:03:40.800 but applying them to other roots this time: simplifying: a radical expression is simplified if the index is as small as possible, 00:03:40.800 --> 00:03:47.200 the radicand contains no nth powers and no fractions, and no radicals are in the denominator. 00:03:47.200 --> 00:03:59.800 Let's go ahead and look at this part: the radicand contains no nth powers. 00:03:59.800 --> 00:04:08.100 Starting out with square roots, just to illustrate this: if you have something like the square root of 18, 00:04:08.100 --> 00:04:16.800 this can be rewritten, using the product property, as 9 (which is 3²) times 2. 00:04:16.800 --> 00:04:27.800 Well, since the root here, the index, is 2, here I have n = 2. 00:04:27.800 --> 00:04:33.100 I look in the radicand, and I have an nth power; I have 2. 00:04:33.100 --> 00:04:42.400 So, this could be further simplified to say "the square root of 3², times the square root of 2, equals 3√2." 00:04:42.400 --> 00:04:47.300 We did this earlier on; but we just restricted this discussion to square roots. 00:04:47.300 --> 00:04:54.300 I could look at a more complicated example, using an index of 3, looking for the cube root. 00:04:54.300 --> 00:05:03.400 If I had something like the cube root of 27, times x⁷: this is not in simplest form, 00:05:03.400 --> 00:05:18.200 because I have an index of 3, and I could rewrite this as 3³...well, x⁷ is equal to x⁶ times x. 00:05:18.200 --> 00:05:32.600 And x⁶ is equal to (x²)³; so if I rewrite this as x⁶(x), 00:05:32.600 --> 00:05:41.500 I could then go on and write this as the cube root of 3³, 00:05:41.500 --> 00:05:49.500 times...since x⁶ is equal to...(x²)³, times x. 00:05:49.500 --> 00:06:02.100 OK, so now, I can see that this was not in simplest form, because I have elements here in the radicand that were raised to the n power. 00:06:02.100 --> 00:06:09.300 So now, what I can do is say, "OK, this 3 essentially cancels out that 3; and I end up with just a 3." 00:06:09.300 --> 00:06:16.200 The cube root of 3³, of 27, is 3; the cube root of x², cubed, is x². 00:06:16.200 --> 00:06:20.600 And that just leaves an x behind, like this. 00:06:20.600 --> 00:06:25.400 OK, so simplest form means that the radicand contains no nth powers. 00:06:25.400 --> 00:06:33.000 Look at the index and make sure that you can't factor out something that would be raised to that same power. 00:06:33.000 --> 00:06:44.000 Also, the radicand cannot contain fractions--no fractions if it is in simplest form. 00:06:44.000 --> 00:06:54.200 Something such as the fourth root of x, over 2z, is not in simplest form, 00:06:54.200 --> 00:07:01.800 because there is a fraction under this fourth root sign. 00:07:01.800 --> 00:07:09.200 And we can use the quotient property to simplify this; and we will talk in a few minutes 00:07:09.200 --> 00:07:15.900 about how you go about getting rid of fractions that are under the radical sign. 00:07:15.900 --> 00:07:26.800 OK, the other thing is: no radicals in the denominator. 00:07:26.800 --> 00:07:35.500 So, if I have something such as 3y divided by the cube root of 2y, this is also not in simplest form. 00:07:35.500 --> 00:07:44.500 And again, we are going to talk about how to get rid of radicals in the denominator, how to go about simplifying those, in just a second. 00:07:44.500 --> 00:07:49.000 So, the index is as small as possible; that is what we are going to be working with (this is usually not an issue; 00:07:49.000 --> 00:07:54.200 usually the index is as small as it can be); the radicand contains no nth powers (so if there is an 00:07:54.200 --> 00:08:02.600 nth power inside, as part of that radicand, we need to take the root of that and remove it from under the radical; 00:08:02.600 --> 00:08:10.500 the same here with the cube root; in addition, the radicand contains no fractions (so something like this 00:08:10.500 --> 00:08:18.200 is not in simplest form); and there are no radicals in the denominator (so this is also not in simplest form). 00:08:18.200 --> 00:08:22.200 You can go through this checklist in your mind, when you think you are done simplifying, 00:08:22.200 --> 00:08:27.800 and make sure that the expression you are working with meets these conditions. 00:08:27.800 --> 00:08:36.400 OK, so I mentioned that you cannot have radicals in the denominator if something is going to be in simplest form. 00:08:36.400 --> 00:08:41.900 So, getting rid of radicals in the denominator is known as rationalizing the denominator. 00:08:41.900 --> 00:08:44.800 So, now we are going to talk about rationalizing denominators. 00:08:44.800 --> 00:08:51.800 And we are going to start out just talking about when we are working with a monomial--when we have a radical in the denominator that is a monomial. 00:08:51.800 --> 00:09:03.300 So, to eliminate radicals in the denominator, multiply the numerator and the denominator by a quantity, so that the radicand is an nth power. 00:09:03.300 --> 00:09:18.100 What does this mean? 2 divided by √5xy: what I want to do is make this radicand (5xy)², 00:09:18.100 --> 00:09:38.700 because here, n equals 2; so I want to get 5xy, squared, as the radicand. 00:09:38.700 --> 00:09:48.700 OK, so in order to do that, what I need to do is multiply the numerator and the denominator by the square root of 5xy. 00:09:48.700 --> 00:10:01.600 This is going to give me...2 divided by the square root of 5xy, times the square root of 5xy, divided by the square root of 5xy. 00:10:01.600 --> 00:10:12.500 Now, I am allowed to do that, because this is just 1; these would cancel out and give me 1, and I am allowed to multiply this by 1. 00:10:12.500 --> 00:10:22.000 So, the numerator is 2 times √(5xy), divided by...looking at the product property: 00:10:22.000 --> 00:10:25.700 the product property tells me that, if I multiply these two, 00:10:25.700 --> 00:10:46.800 I am going to end up with √(5xy times 5xy), which equals √(5xy²). 00:10:46.800 --> 00:10:58.600 So, when I have the index the same as the power that the radicand is raised to, I can just eliminate that radical sign, and then eliminate this power. 00:10:58.600 --> 00:11:06.800 So, I have 2 here, really, although it is not written out, and a 2 here; so I can get rid of both of those and get rid of the radical sign. 00:11:06.800 --> 00:11:14.700 So, this is going to equal 2√5xy, divided by 5xy. 00:11:14.700 --> 00:11:19.100 Now, this is in simplest form, because I no longer have a radical in the denominator. 00:11:19.100 --> 00:11:30.800 And I achieved that by multiplying both the numerator and the denominator by the square root of 5xy, so that I ended up with √(5xy²). 00:11:30.800 --> 00:11:33.800 And I could take the square root of that to just get 5xy. 00:11:33.800 --> 00:11:40.800 But you need to make sure that you multiply both the numerator and the denominator by the same term, 00:11:40.800 --> 00:11:48.100 so that you are actually just really multiplying this entire thing by 1. 00:11:48.100 --> 00:11:55.900 OK, if the expression in the denominator is a radical, but it is a binomial, you need to use a different technique. 00:11:55.900 --> 00:12:00.700 We just talked about rationalizing a denominator when the radical was the monomial. 00:12:00.700 --> 00:12:07.600 But if you have a binomial, then what you need to do is work with a conjugate. 00:12:07.600 --> 00:12:11.100 So, first let's just review what a conjugate is. 00:12:11.100 --> 00:12:21.900 Conjugates are the sum and difference of two terms--not even worrying about the radical sign right now. 00:12:21.900 --> 00:12:37.500 It is something like a + b and a - b; but here, we are working with radicals, so it could be something like √2 + 1 and √2 - 1. 00:12:37.500 --> 00:12:48.800 So, these two are conjugates--the same numbers; the same radical sign; the only difference is that this one is positive; this one is negative. 00:12:48.800 --> 00:13:01.800 OK, so these radical expressions, a√b and c√d and a√b - c√d--the only difference here is in the sign. 00:13:01.800 --> 00:13:12.100 These are called conjugates; so they can be used to rationalize denominators that are binomials. 00:13:12.100 --> 00:13:30.400 For example, if I have 5 divided by the square root of 7 minus 3, well, √7 - 3...the conjugate of that would be √7 + 3. 00:13:30.400 --> 00:13:34.800 So, these two are conjugates; this is a conjugate pair. 00:13:34.800 --> 00:13:52.100 So, what I need to do is multiply 5, divided by the square root of 7, minus 3, times √7 + 3, over √7 + 3. 00:13:52.100 --> 00:14:02.400 Now, since the numerator and the denominator are the same, I am really just multiplying by 1; so again, this is allowed. 00:14:02.400 --> 00:14:21.200 All right, so in the numerator, this gives me...using the distributive property...5√7, plus 5 times 3, which is 15. 00:14:21.200 --> 00:14:26.400 The denominator: recall that here, if you look at what I am doing, it is multiplying a sum and a difference. 00:14:26.400 --> 00:14:37.200 So, if I had a + b times a - b, this is going to end up giving me a²...the outer term is -ab; 00:14:37.200 --> 00:14:45.800 the inner term is positive ab; so that is going to cancel out, and then I am going to get -b². 00:14:45.800 --> 00:14:49.800 b times b is b², but I have a negative sign in front of it. 00:14:49.800 --> 00:14:59.100 So, I am going to end up with a² - b²; and in this case, multiplying √7 - 3 and √7 + 3, 00:14:59.100 --> 00:15:06.100 a equals √7 in this situation, and b equals 3; so this is going to give me 00:15:06.100 --> 00:15:20.200 (√7)² - (I am using this format) 3², which equals 5√7 + 15. 00:15:20.200 --> 00:15:31.900 OK, I have (√7)², √7 times √7; it is just going to give me 7. 00:15:31.900 --> 00:15:44.400 Minus 3² (which is 9) is going to give me 5√7 + 15; 7 - 9 is -2, and I can put that negative out in front. 00:15:44.400 --> 00:15:49.100 So, this is going to give me 5√7 + 15, divided by 2. 00:15:49.100 --> 00:15:51.700 And now, the radical is gone from the denominator. 00:15:51.700 --> 00:16:01.100 So, I have rationalized the denominator when I had a radical, and I had a situation where it was part of a binomial. 00:16:01.100 --> 00:16:02.900 I had a denominator that was a binomial. 00:16:02.900 --> 00:16:13.400 And I did that by multiplying both the numerator and the denominator by the conjugate of the denominator. 00:16:13.400 --> 00:16:20.200 Another review of a concept you may have learned earlier on, which is adding and subtracting radicals: 00:16:20.200 --> 00:16:25.400 radicals are like radicals if they have the same index and the same radicand. 00:16:25.400 --> 00:16:37.200 For example, 4 times the cube root of 5x, plus 2 times the cube root of 5x: 00:16:37.200 --> 00:16:46.100 recall that this is the index; here, the index is 3 and the radicand is 5x; here the index is 3 and the radicand is 5x. 00:16:46.100 --> 00:16:57.600 So, I am going to use the distributive property, and I am going to say, "OK, I have the same; I can pull out this cube root of 5x." 00:16:57.600 --> 00:17:14.600 And that leaves behind 4 + 2; so this becomes the cube root of 5x times 6, or 6 times the cube root of 5x. 00:17:14.600 --> 00:17:23.400 All I did is add 4 and 2; and you can really just look at this as a variable, almost, with the radical. 00:17:23.400 --> 00:17:35.100 If I had given you something like 4y + 2y, that equals 6y; and here, we are going to let y equal the cube root of 5x. 00:17:35.100 --> 00:17:38.500 You could just look at it this way: that this whole thing is like a variable. 00:17:38.500 --> 00:17:43.100 And you can add these two together, but the variables are the same, 00:17:43.100 --> 00:17:50.100 because it is just saying 4 y's and 2 y's equal 6 y's, and that is the same idea here. 00:17:50.100 --> 00:18:01.300 OK, subtraction: the same thing--you just have to be careful (as always, when you are working with subtraction) with the signs. 00:18:01.300 --> 00:18:16.700 So, if I am subtracting something like 5 times the fourth root of 7yz, minus 3 times the fourth root of 7yz, 00:18:16.700 --> 00:18:22.800 I check and see that I have the same index (which is 4) and the same radicand (7yz). 00:18:22.800 --> 00:18:39.300 So, this becomes pulling out the same factor, which is the fourth root of 7yz, leaving behind 5 - 3. 00:18:39.300 --> 00:18:51.000 This is going to give me the fourth root of 7yz times 2, or I am rewriting it as 2 times the fourth root of 7yz. 00:18:51.000 --> 00:18:54.900 So, adding and subtracting like radicals is pretty straightforward. 00:18:54.900 --> 00:19:04.600 Just make sure that you check and make sure that the index numbers are the same, and the radicands are the same, before you try to combine radicals. 00:19:04.600 --> 00:19:09.800 Multiplying: with multiplying radicals, we are going to use the product property. 00:19:09.800 --> 00:19:15.000 And if we are going to multiply sums or differences of radicals, we will be using the distributive property. 00:19:15.000 --> 00:19:20.400 So, let's just start out with multiplying two monomials that involve radicals, 00:19:20.400 --> 00:19:31.200 the fifth root of 2x³ times the fifth root of x². 00:19:31.200 --> 00:19:40.300 Well, the product property, recall, tells me that the square root of ab equals the square root of a, times the square root of b. 00:19:40.300 --> 00:19:46.900 So, what I am doing down here, instead of going from left to right--I am going from right to left. 00:19:46.900 --> 00:19:51.900 I already have these two split up, but the product property tells me I can combine them. 00:19:51.900 --> 00:20:02.500 So, this would actually be the fifth root of 2x³, times x². 00:20:02.500 --> 00:20:14.800 Recall that, if you are multiplying exponents with a like base, then here, I can just add these exponents. 00:20:14.800 --> 00:20:24.000 So, this is going to give me 2 times x⁵. 00:20:24.000 --> 00:20:36.800 Now, I see that what I have (using the product property again) is the fifth root of 2, times the fifth root of x⁵, 00:20:36.800 --> 00:20:43.400 which equals...well, the fifth root of x⁵ is just x, times the fifth root of 2. 00:20:43.400 --> 00:20:51.600 So, you can see how multiplication and using the product property allowed me to actually simplify this. 00:20:51.600 --> 00:20:55.500 First, I used the product property to multiply these two together. 00:20:55.500 --> 00:21:01.800 Then, I used my property of exponents that says I add the exponents, since there are like bases here. 00:21:01.800 --> 00:21:06.700 That gave me 2x⁵; and I saw that this is not in simplest form, 00:21:06.700 --> 00:21:13.900 because I have a radicand that contains the nth power, the fifth power. 00:21:13.900 --> 00:21:15.900 So then, I further simplified. 00:21:15.900 --> 00:21:22.800 OK, so that is if you have monomials; now, for multiplying sums or differences of radicals, we need to use the distributive property. 00:21:22.800 --> 00:21:33.900 For example, if I am multiplying 2 times the square root of 3x, plus the square root of 2, times the square root of x, 00:21:33.900 --> 00:21:40.400 minus 2, times the square root of 5, we are going to use FOIL. 00:21:40.400 --> 00:21:45.500 The first terms (multiplying the first terms, because I am multiplying two binomials--FOIL--First terms): 00:21:45.500 --> 00:21:53.600 that is 2 times the square root of 3x, times the square root of x. 00:21:53.600 --> 00:22:04.600 Plus the outer terms--that is 2√3x, times -2√5. 00:22:04.600 --> 00:22:12.200 Inner terms are √2 times √x. 00:22:12.200 --> 00:22:19.300 And finally, the last terms are √2 times -2√5. 00:22:19.300 --> 00:22:27.000 OK, using the product property right here tells me that √a times √b is √ab. 00:22:27.000 --> 00:22:37.800 So, I am going to apply that here to get 2 times 3x times x, plus 2 times -2...that is actually going to give me a -4; 00:22:37.800 --> 00:22:53.600 so I am going to rewrite this as -4; √3x times 5...3 times 5x, plus the square root of 2x. 00:22:53.600 --> 00:23:00.500 And then here, I have a -2 out in front; and then, that is the square root of 2 times 5. 00:23:00.500 --> 00:23:16.400 I am doing some simplification: this equals 2 times 3x², minus 4 times √15x, plus √2x, minus 2√10. 00:23:16.400 --> 00:23:24.500 And I see here that I am not quite done yet, because I have an index of 2, and my radicand contains something to the second power. 00:23:24.500 --> 00:23:35.900 So, I can pull this out, and I need to remember to use absolute value bars, because this was an even index, 00:23:35.900 --> 00:23:47.400 and when I took that root of this, I ended up with an odd power, 1, so I need to use absolute values. 00:23:47.400 --> 00:23:53.100 And I can't simplify any further, because I can't combine these, since they are not like radicals. 00:23:53.100 --> 00:23:58.700 They are to the same powers, but none of them have the same radicands, so I can't add or subtract them. 00:23:58.700 --> 00:24:04.900 So again, multiplication with sums or differences of radicals--you just use the distributive property, 00:24:04.900 --> 00:24:12.500 like we have earlier on, when working with numbers or variables. 00:24:12.500 --> 00:24:24.900 OK, in this first example, we are going to simplify this expression; and it is the fifth root of x⁸y⁹z⁵, divided by 243. 00:24:24.900 --> 00:24:32.700 So, I am going to use the quotient property, because I know that this equals, according to the quotient property, 00:24:32.700 --> 00:24:44.000 the fifth root of x⁸y⁹z⁵, all divided by the fifth root of 243. 00:24:44.000 --> 00:24:52.500 So, recall that, in order to be in simplest form, a radical expression needs to have an index that is as small as possible; 00:24:52.500 --> 00:25:09.100 no nth powers; no fractions under the radical; and no radicals in the denominator. 00:25:09.100 --> 00:25:20.200 So, you should be familiar with these rules. 00:25:20.200 --> 00:25:28.800 What we have is: when we started out, we knew it wasn't in simplest form, because I did have a fraction under that radical. 00:25:28.800 --> 00:25:33.700 I then took care of that by using the quotient property. 00:25:33.700 --> 00:25:39.300 I no longer have a big radical sign where I have this fraction under it; I split it up. 00:25:39.300 --> 00:25:43.700 The only problem is that I now have a radical in the denominator. 00:25:43.700 --> 00:25:47.800 So, this is still not in simplest form. 00:25:47.800 --> 00:25:54.400 In addition, I also have some nth powers under here; when I work on this, 00:25:54.400 --> 00:26:01.400 I will see that there are some terms here that are to the fifth power. 00:26:01.400 --> 00:26:12.300 So, I can rewrite this as x⁵ times x³, because these have like bases, so I add the exponents. 00:26:12.300 --> 00:26:22.400 That would give me x⁸ back, so I can see here that I have n = 5, and the radicand contains some fifth powers. 00:26:22.400 --> 00:26:33.500 y⁹ would be y⁵ times y⁴, because I would add these to get 9 back; and then leave z as it is, z⁵. 00:26:33.500 --> 00:26:44.300 243 is not totally obvious, but it turns out, if you work this out, that 3 to the fifth power is 243. 00:26:44.300 --> 00:26:49.900 So, I also have a fifth power as part of the radicand in the denominator. 00:26:49.900 --> 00:26:56.000 OK, I am going to use the product property to rewrite this with my fifth powers all together here: 00:26:56.000 --> 00:27:20.700 x⁵y⁵z⁵ times what is left over (that is the fifth root of x³y⁵). 00:27:20.700 --> 00:27:28.900 OK, what this gives me is the fifth root of these; well, these are all to the fifth power. 00:27:28.900 --> 00:27:38.400 So, I simply remove the radical, get rid of the n, and get rid of this power to get x; the same with y and z. 00:27:38.400 --> 00:27:43.000 Since this is odd, I don't have to worry about absolute value bars. 00:27:43.000 --> 00:27:47.500 We only worry about that when the index is even. 00:27:47.500 --> 00:27:55.000 And this is times the fifth root of x³; this should actually be y⁴ right here, 00:27:55.000 --> 00:28:05.900 and y⁵ here and y⁴ here, divided by...well, this is the fifth root, and this is raised to the fifth power. 00:28:05.900 --> 00:28:22.100 So, that just becomes 3; or I could rewrite this as xyz divided by 3, all that times the fifth root of xy⁴. 00:28:22.100 --> 00:28:24.700 So, I double-check: is this in simplest form? 00:28:24.700 --> 00:28:33.200 There are no nth powers in the radicand; what I have left is xy⁴; there are no fifth powers here. 00:28:33.200 --> 00:28:39.700 There are no fractions under this radical sign; there are no radicals in the denominator. 00:28:39.700 --> 00:28:45.000 So, this is in simplest form. 00:28:45.000 --> 00:28:53.800 Here, we are asked to add and subtract some square roots; recall, though, that you can only add or subtract radicals 00:28:53.800 --> 00:28:59.400 if they have the same index (which these all do) and the same radicand (which they don't). 00:28:59.400 --> 00:29:04.400 However, they are not in simplest form yet; it is important to always simplify first. 00:29:04.400 --> 00:29:11.700 The problem is that I have some perfect squares left here as part of the radicand that I could pull out. 00:29:11.700 --> 00:29:16.300 So, I am going to rewrite this with the perfect squares factored out. 00:29:16.300 --> 00:29:28.900 And I am going to use the product property; 24 is 4 times 6; 48 is 16 times 3, so I have a perfect square; 00:29:28.900 --> 00:29:37.600 54 is 9 times 6, so that is another perfect square; and then, 75 is 25 times 3. 00:29:37.600 --> 00:29:46.000 According to the product property, the square root of ab equals the square root of a times the square root of b. 00:29:46.000 --> 00:30:02.500 So, I can rewrite this as √4 times √6, minus √16 times √3, plus √9 times √6, minus √25 times √3. 00:30:02.500 --> 00:30:06.000 And as you get better at this, you might not need to write out every step. 00:30:06.000 --> 00:30:08.800 But for now, it is a good idea, just to keep track. 00:30:08.800 --> 00:30:25.400 This is going to give me 2√6, minus √16, which is 4, √3; plus √9, which is 3, √6; minus √25, which is 5, √3. 00:30:25.400 --> 00:30:32.500 Now, I am looking, and I actually do have some like radicals that I can add, because they all have the same index; 00:30:32.500 --> 00:30:47.400 and now I see that these two (let's rewrite it like this): 2√3 + 3√6, have the same radicand. 00:30:47.400 --> 00:30:55.200 And then, I have -4√3 - 5√3. 00:30:55.200 --> 00:31:05.100 In this situation, what I am going to do is add the 2 and the 3, and this is going to give me 5√6. 00:31:05.100 --> 00:31:13.400 You are looking at this the same way that you would a variable: if I had 2y and 3y, it would become 5y. 00:31:13.400 --> 00:31:29.300 Plus...this is going to be -4 and -5; that is going to be -9, and then √3, which is 5√6 - 9√3. 00:31:29.300 --> 00:31:36.800 So, this is now in simplest form; and when I looked at this, it first looked like I could not combine these. 00:31:36.800 --> 00:31:42.400 But when I went about my checklist of how to simplify, I didn't have to worry about radicals in the denominator 00:31:42.400 --> 00:31:49.200 or fractions under the radical sign, but I did have to get rid of the perfect squares that were part of the radicand. 00:31:49.200 --> 00:31:52.600 And I used the product property to achieve that. 00:31:52.600 --> 00:32:01.800 Once I did that, I saw that I actually could combine some of these radicals to get this simplest form. 00:32:01.800 --> 00:32:06.700 OK, simplify: this time, I am asked to multiply two binomials. 00:32:06.700 --> 00:32:12.100 And I am going to use FOIL, just like I normally would. 00:32:12.100 --> 00:32:22.300 I am rewriting this here; use FOIL just as though you are multiplying any other two binomials. 00:32:22.300 --> 00:32:30.200 So, this is going to give me 2√3 times the other first term, which is 6√3, 00:32:30.200 --> 00:32:38.400 plus 2√3 (the outer terms) times 2√5. 00:32:38.400 --> 00:32:54.700 Now the inner terms are -4√5 times 6√3; and then the last terms are -4√5 times 2√5. 00:32:54.700 --> 00:32:59.000 OK, I am making sure that I have everything correct...inner, and then last. 00:32:59.000 --> 00:33:12.400 OK, we can use the product property; and the product property tells me that √ab = √a times √b. 00:33:12.400 --> 00:33:18.200 And I am actually moving from right to left here, because these are separated, and I want to put them together. 00:33:18.200 --> 00:33:34.900 So, this is going to give me 2 times 6, √3 times 3, plus 2 times 2, and then this is √3 times 5, 00:33:34.900 --> 00:33:49.000 plus -4 times 6, and this is √5 times 3, plus -4 times 2, times √5 times 5. 00:33:49.000 --> 00:33:57.900 This gives me 12; and I could rewrite this as 3² + 4; this is 15. 00:33:57.900 --> 00:34:09.100 -4 times 6 is going to give me -24√15; -4 and 2 is going to give me -8; and I can write this as 5². 00:34:09.100 --> 00:34:14.900 OK, I am not done simplifying yet, because recall that I look at the index; it is 2; 00:34:14.900 --> 00:34:21.700 and I see that I do have a term here and here that are to the power of 2. 00:34:21.700 --> 00:34:29.000 I can take 3²; that square root is just going to be 3; so this gives me 12 times 3. 00:34:29.000 --> 00:34:36.000 This does not have any perfect squares within it as factors, so I leave it alone. 00:34:36.000 --> 00:34:44.700 The same with this term; here, I do have 5², so the square root of 5 squared is just 5. 00:34:44.700 --> 00:34:56.300 Continuing to simplify: 12 times 3 is 36; -8 times 5 is -40. 00:34:56.300 --> 00:35:05.900 I can combine these two, 36 - 40; that is going to give me -4. 00:35:05.900 --> 00:35:16.600 I also can combine these two, because these are the same index, 2, and they are the same radicand. 00:35:16.600 --> 00:35:28.500 So, this would be the same as 4 - 24 times √15, 00:35:28.500 --> 00:35:39.700 which is going to give me -4; and then 4 - 24 is just going to give me -20√15. 00:35:39.700 --> 00:35:50.900 So, the simplified expression here is this; and I know it is in simplest form, because I don't have any fractions under the radical sign. 00:35:50.900 --> 00:35:59.800 There are no radicals in the denominator; and I don't have any nth powers here; I don't have any perfect squares under here. 00:35:59.800 --> 00:36:06.700 So, I first multiplied these two binomials out, using the distributive property. 00:36:06.700 --> 00:36:16.200 I got down to here; then I took my perfect squares; I took the square root of this number, 9, which is 3. 00:36:16.200 --> 00:36:20.900 I took this square root of 5², which is 5. 00:36:20.900 --> 00:36:34.500 I did some more simplifying, and then combined these two radicals that had like radicands and had the same index. 00:36:34.500 --> 00:36:40.000 OK, Example 4: Thinking about my rules, is this is simplest form? 00:36:40.000 --> 00:36:48.400 No; it doesn't have any fractions under the radical; however, there is a radical in the denominator. 00:36:48.400 --> 00:36:54.200 So, a radical expression is not in simplest form if there is a radical in the denominator. 00:36:54.200 --> 00:37:03.400 Recall that, to simplify a radical binomial expression, you multiply both the numerator and the denominator by the conjugate of the denominator. 00:37:03.400 --> 00:37:12.200 So here, I have 2 + √3; the conjugate of that is going to be 2 - √3. 00:37:12.200 --> 00:37:18.600 So, these are conjugates; this is a conjugate pair. 00:37:18.600 --> 00:37:35.100 I am going to take 4 - √3, divided by 2 + √3; and I am going to multiply that times this conjugate, 2 - √3. 00:37:35.100 --> 00:37:39.400 This is just the same as multiplying this by 1. 00:37:39.400 --> 00:37:46.500 I am going to have to use the distributive property, because I am multiplying these binomials. 00:37:46.500 --> 00:37:49.500 So here, I am just going to have to use FOIL, as usual. 00:37:49.500 --> 00:37:59.100 The first two terms are going to give me 4 times 2; the outer is going to give me 4 times -√3. 00:37:59.100 --> 00:38:14.400 The inner two terms--that is -√3 times 2; and then, the last two terms are -√3 times -√3. 00:38:14.400 --> 00:38:23.500 The denominator is a little bit easier, because this denominator is in the form (a + b) (a - b). 00:38:23.500 --> 00:38:27.300 It is the product of a sum and a difference, which gives me a² - b². 00:38:27.300 --> 00:38:45.100 Here, a = 2, and b = √3; so that is going to give me a², which is 2², minus (√3)². 00:38:45.100 --> 00:39:06.400 OK, simplifying: 4 times 2 is 8; this is 4 times -1, so that is -4√3; this is -1, essentially, in front of here, times 2 is -2√3. 00:39:06.400 --> 00:39:16.500 Here, I have a negative and a negative; that is going to give me a positive, so it is going to be + √3; and that is squared. 00:39:16.500 --> 00:39:28.200 OK, all divided by...2² is 4; minus...well, the square root of 3 squared is just 3. 00:39:28.200 --> 00:39:43.200 OK, so this gives me 8 - 4√3 - 2√3. 00:39:43.200 --> 00:39:51.200 Well, this √3 squared is also 3, so I am going to change that to a 3...divided by 4 - 3, which is 1... 00:39:51.200 --> 00:40:06.600 so I can just not write that 1; and here I have 8 + 3; that is 11; I also see that I have -4√3 and -2√3. 00:40:06.600 --> 00:40:13.800 Since these have the same index and the same radicand, I can combine these two to get -6√3. 00:40:13.800 --> 00:40:19.600 So, we started out with something not in simplest form, because it had a radical in the denominator. 00:40:19.600 --> 00:40:26.900 And since that was a binomial, I multiplied both the numerator and the denominator by the conjugate of the denominator. 00:40:26.900 --> 00:40:34.100 I got this whole thing; I then just continued to simplify. 00:40:34.100 --> 00:40:43.900 And when I got to here, I saw that I had two radicals that could be combined to give me -6√3. 00:40:43.900 --> 00:40:45.900 And then, I combined my constants. 00:40:45.900 --> 00:40:50.300 And this is now in simplest form, because there are no radicals in the denominator; 00:40:50.300 --> 00:40:57.800 there are no nth powers in this radicand (no perfect squares, in this case); 00:40:57.800 --> 00:41:06.200 and no fractions under the radical sign; and the index is the smallest power that it can be. 00:41:06.200 --> 00:41:11.000 That concludes this session of Educator.com; thanks for visiting!