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Rationalizing Denominators and Simplifying Radicals with Variables (Square Roots)

Added Apr 03, 2020 •

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flower everyone this is Miss Kilgore again and geometry Jane welcoming need to learning live and also algebra Annie today welcoming you to learning live now we're going to do a joint lesson today between Algebra 2 and geometry because Algebra 2 is simplifying radicals right now and geometry is going to need to simplify some radical for their Pythagorean theorem Algebra 2 we will move a little bit farther than this and we talked about it I'm in class and we'll move on to some cube roots and some fourth roots geometry you don't have to worry about that right now we'll get to it when you get towels for two but for right now we can use the same concept for just a few is going to have some preliminary exercises here to do on ixl.com feeling just with and square root simplify fractional radicals and swelling and deal a little bit with something we call rationalizing the denominator but this is for both classes and then we'll Branch off to just individual lessons for both in the next day or so for now though we are going to do some simplification of radical and we may be repeating here just a little bit a little repetition generally does hurt most of it I'm going to starve here with the simplified and I'm going to start with some simple sign of the variables raised to X and a low price of numbers back in in but I started here with the square root of x squared and of course we know if there is no number there in that little crook of that radical that is under said to Dad square root that we have a square root of x square and all that that square root and that's where they're going to do to each other is to cancel each other out so the square root of x squared is simply at orex today first if you prefer now when I come down here to the square root of x to the eighth remembers were asking for the square root of something we are simply asking what time is it Dale will give us at to the A and we want something that will X it fail so we're going to have to remember a rules of exponents here and when multiplied we add our expense so what we're going to have tears in the square root of x to the 8th is just going to be x to the 4th because x to the fourth High Times at 2 4th is X to the A and that's their whole reason for that now I want you to notice something here when we did notice the trip whenever I hear that your power you're taking this to this is the square root is this an index of to notice that your exponent in here is divisible by 2 8 / 2 is 4 we like it when our exponent is divisible by the end of that because what happens is all we have to do is just divide it so if I give you the square roof in of x 8/22 22 is divisible by 2 so the square root of x to the 22nd is just today if that even wear in good shape if we're taking a square root so we can even say hey what's the square root of x CD and the square root of x to the stick put simply be at today and where we wanted to take the square a Samsung to an exponent in the exponent is not divisible by 6 so let's do a lot if here and let's say that we want to take the square root avexa day 1515 is not divisible by X2 is it not so well yeah it is a whole number is nothing is divisible by something else we're going to have to rewrite this one and then we took the square root of what we could that what we need to do here is to you their exponent rules and think about x to the 15th as X something what if we try to the 14th x x to the first that room when we multiply we I did Stannis 14th plus one would give us a text to the 15th so we're in good shape there now we're ready to take the square root of a can so what is the square root of it the 14th going to be we're going to divide 14 by 2 and that's going to give up at 2 the Fanta put it outside and then how about the square root of x to the first one who is it that we're going to leave it at your nannies and we end up then with X to the 7th on the √ at believe it or not what's the looks of that but that is the simplified version of the square root of x to the 15th now notice I did put that to the first of course you don't have to you can just leave it at that but also here's a little clue for you when that at spine that's left underneath the radical is less than your index here one is less than 2 and we're thrilled we have done all day we can do it hey any questions there think about that front and let that sort of thinking and remember send me your questions by email I was trying to how about √ at cd27 27 is not divisible by 2 is it 2 is there in debt the left rear to the 27th so remember I'm trying to get an exponent left underneath that is less than 2 but I think I'm going to have a next to the first in there and then I'm going to have an x to the 26 because when I multiply add my exponents the 26 + 1 is 27 and then we are going to take the square root of 4 we can so what's the square root of x in the 26th we divide 26 by to so that's a x to XIII it goes outside and then we cannot take the square connect to the first that we had access to the 13 * √ at now for algebra two listeners remember we're going to go a little farther than that and we have somewhat already and we're going to change that to maybe not just a square root but it's cube root of 4 thread will do that on that a separate video and do a little bit more practice with that probably tomorrow for geometry this is all you need at the moment any questions all right let's move a little farther than that sorry about that noise I hear in my legal pad off there and let's put everything said so far together and let's do one with four we simplify one with numbers and we got some variables and some exponents under there because you could very well have one ideas on your exercises here after this let's Okay so we are going to do then the square root of 24 WW2 and we might have to do several different little pieces here so let's think about are 24 first of all 24 is not on my list perfect square if I put it in my my calculator is going to give me four points and I think and I don't want a decimal here I want a simplified version so let's rewrite 24 as a perk square of 4 times yes you could say 2 x 12 it just may not be very helpful to us and I'm going to leave it like that for a while I'm going to I'm going to rewrite everything simplify 8 to the 7th 7 is not divisible by 2 so let's go ahead and let's rewrite that how about is X to the sixth time at 2 1st number for one of them I want to get that less less than that I'm taking mmw to 4th that's an even it it's okay the way it is it will work just a Zia so let's leave it as W to the port a what's the square root of 4 YouTube can I take the square root of 50 who is Donna latest underneath my radical Leave myself a little bit of room here because I think I'm going to have some more thing because there was no variable solid stuff a little hole K what's the square root of it big sister visible boxes that only works for the exponent so make sure you only do that if you have an answer for the square root of x to the third it'll come outside can I take the square root of x for the first can't do it one is less than to talk keep my ex underneath and then we're down to the W what's the √ WW2 4th for is divisible boxes at 4 W square knot yeah it is simplified version and you wouldn't have had that radical quite so long I got a little carried away there but we have simplify the square root of 24 x 10 to the 4 W squared times the square root of 6 ax now we would consider this simplified whether we like the looks of it that would be our prayer for and that would be what would be considered simplified in math math any Quest how do I say just a little bit farther and say okay what would happen if maybe we fraction underneath the radical could we simplify it we're going to start with a couple of things here and here again we are going to be simplified hairless tiger look at the first one and for the first when we're going to have the square root of 49/1 Android 21 one more simplify Nanny kind of radical folks there are a couple of couple of things that we don't want never want to leave a fraction underneath the radical that's considered simplify a radical in the denominator of a fraction either it's okay to rider in the top but not in the box and I look at their one thing we could do it's perfectly acceptable to split days and make it the square root of 49 over the square root of 121 and then we can work with them separately now I think we lucked out on this one because we ended up with the square root of 49 + 7 the square root of 121 is a perfect square and that gives us 11 absolutely not wrong with 7/11 as an answer 7/11 is a nice fraction Amplified as far as it can go and where. Quest hey now as you might expect they don't all come out quite as nicely as that okay take a look and look what good couple of others here ring with simplifying fractions that one way and I may see it another way and we may start at different places but we may end up at exactly the same plaid but sometimes we what place an order something and it's perfectly legal in case I'm going to try to show you a couple of examples here for you can see pretty might go maybe One Direction I'm not going other okay let's try this but the time that we have the square root of one hundred 25 divided by the square root of 45 now if you look at that we got a couple issues we don't want to leave that √ 45 in the denominator we don't want to leave a radical in the denominator but 45 non-perfect Square so it's not going to be as simple as just saying I'll take the square root of 45 and it'll be whatever I have to work on this one little bit now what I think I'm going to do here they both appear to be divisible by 5 so I can rear I didn't think about it like this. If I want to √ 125 / 45 and then I could do a little mental math or I could use calculator if I wanted to I could even put that in my calculator and hit math in her in her but I think I can do this 125 / 5 is about 25 I believe and in 45/5 is 9 so I'm just going to simplify the fraction first after that have my square root symbol can't just throw it away but now look at how Ashley that should have said that up for me just by simplifying the fraction I got it down to a couple of perfect square and it doesn't nearly as bad as it did now I can rewrite it as the square root of 25 over the square root of 9 square root of 25 going to be 5 isn't it and what's 2 square rooted give me a 3 and I have gone all the way from square root of 125 over square root of 45 all the way to 5 over and I'm done how did that go friend regarding any questions remember let me know if you didn't follow something and all turn on your eyes self exercises remember that there's a little box on each question where it's going to give you a chance if you didn't understand something to click on it and it'll give you an example it may take you a little you may help your score if you'll take just a minute and look at those clues that they give you that remember I'm always available. All right are we ready to move to a couple more problem okay then let's look at just a couple more and we're going to look at one more situation babysitting don't turn what is a perfect square and we took out a deal with take a look at something like the following we're still simplifying right now. Just a little bit not forget it in the screen like I told you I'm still tired of learning my my document camera here so be patient with me we're going to simplified I think this is number 3 through this section and this time we're going to do the square root of 10/15 now you know look at that and remember I always Advocate to look at your problem get yourself appointment hey see what you think you can do see what might be helpful to you so when you look at this if I look at 10:15 or perfect square so I'm not going to be do you see a fraction maybe I can simplify maybe that might help me a little bit in the end I think I'm going to take that. So I'm going to go ahead and reduce my 10/15 by 5 that's going to give me two of her story and I look at it so let me see a whole lot of good did you still got a radical around a fraction and that's absolutely true take a look at it and say we'll find out we do this how about if we make it the square root of 2 over the square root of 3 well that's true that's perfectly legal but it's not simplified yet and this is not simplified cuz I've got a radical in the denominator so this is where we're going to use a process called rationalizing the denominator running out of room they're running into my document camera it's called rationalizing the denominator the denominators but I'm concerned with so here's the easiest why I think to remember that when you are rationalizing the denominator look at what your diarrhea's look at what you don't want their so here I don't want to √ 3 a plan 03 would be great if I had a square root of 11 I wouldn't want a square root of a laugh so I thought we're going to do we are going to * the square root of 3 because what's going to happen is the square root of 3 times the square root of 3 gives us the square root of and I can take the square root of 9 and that gives me a final 3 if I'm off, denominator by the √ 3 guess what I have to do to the name Ryder I have the X design thing so I have to * √ 3 Eminem Riders well and I that because what's happening here is what I X that square root of 3 over square root of 3 what's anything divided by itself took the one isn't it and when I reply by one are you going to change the value of anything no you're going to get exactly the same thing as you started with so that's why I have to multiply the same thing top so that I'm multiplying that whole thing by one okay so as I said whatever you're trying to eliminate if you have a square root on the bottom and you don't want it the most square root of 11 over square root of Atlanta if you have a square root of 5 in the denominator and you need the x * square root of 5 over square root of 5 K Y might stay up here we just have to do the multiplication how much are ear √ 2 * √ 3 + √ 6 you can always multiply two radicals as long as they're the same thing and it's square root of square root you can multiply those numbers underneath but that's going to be square root of 6 Square Root 6 will not simplified or we'll leave it as he has if it would simplify we good much as we could with it that we're going to be left with √ 6/3 resist the temptation folks don't say I'm going to divide 6 by 3 that is not a plane 06 square root sticks that sits is responsible to that square root symbol first okay so you has gone as far as you can go with it you cannot simplify any farther than √ 6 out of three you're done okay radical cannot be divided by whole number everybody with me okay I've got just a couple more problems that I wanted here maybe about 5 that I want to practice with you and I know that's going to put it in the maybe 10 or 12 more minutes but remember you can always stop this when you want to you can always read find it if you need to and go back Hobart okay and I'm trying to be as detailed as possible if I need to be more detailed please let me know alright okay let's try another one here Kayla fry the square root of 121 the square root of 40 now I tell you all a lot different people see different things at different times Okay so when we look at this I may see something and you may have seen something else before but remember we can swap order here sometime as of the first thing that jumps out on me on this one. Hyman Riders a perfect square what's the square root of 121 2011 is at 5:11 in my numerator so I've already dealt with the numerator sign. √ 40 is that a perfect snot is it so we're looking at it without a little bit of an issue here we don't want to have a radical in the denominator and of course the 40 needs to be simplified so we can either simplify the radical and then rationalize if need to be where we can rationalize it right the way we are I'm going to simplify first that's just my preference so I'm going to rain what's the square root of 40 as the square root of 4 * 10 and then I'm going to take the square root of what I can so I have a 11 / square root of 4 what do square root of 10 can't do anything with it can I sell end up with 11 he's me over to on the square root of 10 getting a little closer to where we want to be but what's the problem with it there's still one slight problem that radical in the denominator isn't it so here's what I'm going to do remember we it's whatever radicals in the denominator that you don't want their that's what we're going to * let's * √ 10 that they will just come along for the ride will deal with it in a man √ 10 that were interested in. X square root of 10 over square root of 10 cuz remember we don't want to change the value we just want a * 1 4 * √ 10 / √ 10 ls single working a cat 11 is a whole number square root of 10 as a radical so all we can do is by 11 * 2 over 2 times the square root of 100 square root of 10 times square root of 10 now I'm sitting at staff any Mega stripe 2T on all night that's okay 3 of 11 on the square root of 10 all / 2 x remember to you on the √ 100 means two times the square root of 100 square root of 110 * 2 * 10 is 20 let's look at what we got I think it's Canada Swinney that can they be cancelled no Keenan is underneath that radical it's responsible to the right medical first can't do anything with that 1120 did they have something in common they don't do that so I'm through my square root of 1 1 / square root of 40 has simplify 211 on the square root of 10 / 20 do I have any questions at all Kyle has tried just a couple more here hang in there with me positive finite and take a break come back to it that we've got square root of 50 over square root of 12 not here again I'm doing what sense to me hey what jumped out at me first except for this one I'm going to think about it is square root of 50 / 12 so I can see a fracture the numerator and the denominator divisible box it Tom Hiddleston I might even sit in my calculator if I wanted to hit math in her inner thigh square root with it just putting 50 / 12 and hit math inner inner that 50 is divisible by 2 that gives me a 25 12 is divisible by that you can't see it help me a little bit because now I have square root of 25 over square root of 6 remember I put in every step possible here you do not have to put in all of these except what's the square root of 20 top that's going to be at 5 can I take the square root of can't do it Canna so I'll leave it in the denominator now we have to ask yourself one more question is that possible absolutely not because we can't leave that radical in the denominator so remember we're going to use their process here of rationalizing the Terminator so we are going to * the square root of 6 over the square root of 6 and we keep doing that I'm going to keep circling it because remember we've got to keep it the same value as we started out with Thor x one that we end up being with 5 on the square root of 6 all over the √ 36 and yay-yay √ 36 gives us that we end up as 5 on the square root of 6 over fix now be careful here folks resist the temptation because we look at that when I say hey there's a cyst in the name writer there's a 6 in the denominator Naruto radical sit and a whole number sit are two different creatures they do not give you one so you got to stop with what you got. Right here 5 on the square root of x over  that's as far as we can get how are we doing out there any questions all right we've got just a couple more to go there I am with and we're almost done with this lesson hey we're still simplified and we are on question ever see it so let's go with the square root of 40 what is the square root of 12 now I want to do these last two to put it show you one more little step we can go with here and you can come like the one we just did Florida I want you to see the The Final Answer here okay now for me Doolittle analysis on it can't take the square root of a one of those I can simplify a little bit but I think I'm going to deal with with the diffraction and then start simplified let's go ahead and let's Freeride physicist square root of 40 / 12 and let's reduce 40/12 to 40/12 in your calculator hit math inner inner if you'd like to do and when we do that I think divisible by 4 right so we are going to end up with the square root of 10/3 when you leave is that you may feel great that looks better that remember we got a fraction underneath the radical if you rewrote it you still have a radical in the denominator which we don't so we're going to have to rationalize it so what are we going to multiply top and bottom by I'm going to start it over again on this bottom on this next line weather hey we are going to * √ 3 ever swear read it three whatever it is that I don't want that denominator is what I'm going to x honey the most top of the Square Reader and then when it comes down to it multiply and top and bottom of the same thing being remember so I got the square root of 30 Oliver √ 9/3 * 3 that's a easy mistake to make sometimes √ 30 is what I have on the top look how nicely that √ 9 comes out what's the √ it's just a plain old free isn't it now I did this one because I want to take this opportunity again to remind you that you can never ever ever heard of a square is the square root of 10 never ever ever that party is responsible to that square root December 1st 03 can do nothing else except a pimple or an approximation on your calculator and virtual instruction say simplify you don't want to get an approximation okay alright any questions okay then, we got one last of the prom when your we want to do and I want to show you one last thing that you do your situation a lot I said this time let's try this when we are going to simplify 8 square root of 20 over square root of 8 set a whole number in here with this and I'm just going to kind of bring it along the ride from Moana that's the first thing I'm going to do is I'm going to rewrite it like this I'm going to rewrite it as a 8 times the square root of 20/8 and then I'm going to reduce that fraction so let's go ahead and make this ain't * √ what's that going to give me 20 of the visible body for that'll be 5/2 and that right hey darnold have a tan the square root of 5 over the square root off and all I did was just amplify the square root of 20 over the square root of 8 still got a little bit of a problem here I still got a denominator and she's me Ida nominator that is rather a radical got to deal with that rap so what am I memo father square root of 2 over the square root of them said it would really just multiplying by a 100K in the numerator then we're going to go 8 on the square root of 10 in the denominator we're going to get the square root of 4 and look what's going to happen what's the square root of 4 is that why no regular to and then we have I8 on the square root of 10/2 now we can do a little bit more simplify true or false can I say square root of 10 divided by 2 is the square root of 5 I hope I had a lot of false it's cuz that should be false true or false can I divide 8 by 2 the answer to that one should be true for those are whole numbers that's great you're just reducing the fraction 8/6 lowest for you can never ever ever though divided what's underneath the radical of a whole number on the bottom okay so we're just going to bring that radical along and we are going to end up with 4 on the Square radar all right that's it help you with your next exercise and this is really just being for geometry just an algebra review for what you're getting ready to get in today with and of course with the the algebra 2 I think I may have said that incorrectly for the geometry is an Audi S4 review for your platter and Theorem Algebra 2 this is just sort of a a practice to get you used to thinking of the cube roots in the fourth roots as well and we will do some of that fried to pursue on a pleasant. Let me know if you have questions and keep working hard at what you doing see you next time bye bye

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Rationalizing Denominators and Simplifying Radicals with Variables (Square Roots)

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