in this video we're going to take a look at multiplying and dividing radical expressions when we're multiplying radical expressions what we want to do is take the radicand in other words what's inside the square roots and we can multiply those things together and get that combination that product under the square root and then from there we can do some simplification so let's take a look at this first one what is the only operation that's going on between all that stuff well multiplication it's 3 * the < TK 6X * the < TK of 10x I can multiply those radicans right there the 6X and the 10x to give me the square root of 6 * 10 is 60 and then x * X is X2 okay so then we have that three on the outside don't lose track of that now we can take a look at what's under the square root and see if there's any perfect square factors there if there are we can break it up and do some simplification so let's see hm perfect squares factors of 60 perfect squares four is a perfect square um that goes into 60 4 * 15 um 9 no 16 no 25 no 36 no 49 no 64 too big so we're done okay so four all right so we've got three times we're going to break this thing up to the square < TK of 4 because that's a perfect square Factor times the square < TK of 15 time the square < TK of X2 okay so here I multiplied the radicans to get one square root piece and then I turned around and I broke it up again because what that allows me to do is some simplification I can take the square root of 4 that's two so I'm going to have 3 * 2 * squar of 15 well perfect square factors of 15 nope there aren't any so I just got to bring that along and then I have the square root of X2 well that's just going to be X Okay so then I'm going to clean up and what can I pull together here well I've got 3 * 2 * X all those things that are on the outside of the square root that would be 6X and then inside the square root I have that > of 15 so I have 6X * the > of 15 all right let's take a look at this next one here the rules for multiplying binomials follow whether whether it's a just a regular old number or a radical we still can use the foil method and figure out what we're going to get so let's go ahead and foil these two binomials and then we'll simplify so 4 * 1 first terms is Just 4 then 4 * - < TK 5 is going to be - 4 Square < TK of 5 okay then we've got sare < TK 5 * 1 that would be plus the < TK 5 then < TK 5 * minus the < TK 5 let me just write that out here so we don't lose track of what all is going on so it's minus theun 5 Time theun 5 okay then let's go back and do some simplification four can't do anything with that yet then I've got Min -4 * a > 5 plus theun 5 well remember that we can add add those together because they're like radicals so we have that same square root square root of five so I have Min -4 of those plus one of those so I'd have -3 squ < TK 5 then over here squ of 5 * the < TK of 5 remember we can multiply the radicand so 5 * 5 is 25 so it's minus the square < TK of 25 okay so then I'm going to go back through and simplify anything more that I can so I have four can't do anything else with that yet can't do anything else with the three or minus 3 < TK 5 however minus the square < TK of 25 well that would just be five okay getting a little simpler and now I see something else I can combine I can combine the four and the minus5 so that's going to give me -1 - 3 square root of five all right let's take a look at somewhere we have some division going on here all right in this one there's a couple ways we can approach this remember that if we have a square root over a square root we could rewrite that as a fraction the square root of that fraction 2 over 6 and then simplify inside the square root there and that would be the square root of 13 okay then remember that then I can take the square root of the numerator and the denominator so I have the square < TK of 1 over theare < TK of 3 now the square < TK of one is just 1 over the < TK of 3 okay so then I need to rationalize the denominator remember we can't leave a square root in the denominator so to rationalize what I'm going to do is I'm going to mul multiply by that square root right there so if I multiply by that * < 3 * 3 on top 1 * 3 is just thek 3 all over we've got thek of 3 * the square of three which would be the square root of 9 well the square root of 9 is just three so notice how we end up with whatever was inside there whatever the radicand is when we multiply it by itself so that's that now we could also just take this and rationalize from there let's take a look at what that might look like I'm going to switch colors here just so we can tell the difference so rather than pulling it back into a fraction and simplifying and then taking it back out I'm just going to go ahead and rationalize right here so to do that I would multiply by the square < TK of 6 on the top and the bottom so I have the square < TK of two over over the > of 6 I'm going to multiply by the sare < TK of 6 on the top and the bottom then I have on top the sare < TK of 12 overk 6 * 6 is just 6 okay you might say wait a minute something's not right H but wait square of 12 any perfect square factors yes four will go into 12 so I can break that up into the square < TK of 4 time the square < TK of 3 all over six still square > of 4 is 2 so I have 2 * the < TK 3 over 6 I can simplify this stuff sitting right here divide by two top and bottom so that would give me same thing okay so notice two different methods we could approach that and we end up in the same place the key is making sure that we look look for those perfect square factors things that we can simplify and taking it from there okay let's take a look at this last one here we've got the square < TK of 24 over 4 * the < TK 3 well I need to rationalize that denominator and I also there's a perfect square factor of 24 so I've got a choice to make I can start by simplifying that uh 24 and then worry about rationalizing or I can rationalize and then simplify I'm going to choose to rationalize and then simplify so I've got the I'm going to multiply by the square root of three on the top and the bottom so then I get on top 24 * the < TK 3 would be um 72 > of 72 all over 3 * 3 would be thek 9 which is 3 remember and then we've got 4 * 3 on the bottom I'll just write it like that for now now square of 72 perfect square factors let's see H nine goes in there however if I think about this for a second 9 9 * 8 is 72 8 still has a perfect square Factor so maybe I should try to look for something bigger hm how about 36 36 6 * 2 is 72 ah Perfect all right so I'm going to go over here with that so I have the square < TK of 36 * theare < TK of 2 all over 4 * 3 is just 12 okay then Square 36 is 6 * theun 2 all over 12 then I can do some simplification remember we can divide top and bottom by the same number to simplify just a fraction thinking of that 6 over2 12 there and that would be 1 and that would be two so it' be the square < TK 1 * the < TK of two or just squ > two over two all right multiplying and dividing radical expressions remember that if we have two square roots that are being multiplied we can multiply the radicans and then take whatever we get from there put that under the square root and then break it up and simplify from there foil method still applies even when we have radicals and also we cannot leave a square root in the denominator so we have to do process called rationalizing the denominator to get it out of there and get our our fraction simplified hope this video is helpful keep working hard on your math and I know you can do great stuff