in this video we're going to focus on how to simplify radicals including radicals with variables we're going to also focus on how to add and subtract radicals how to multiply radicals and divide them including how to rationalize the denominator and when to multiply by the conjugate so let's start with simplifying radicals you need to know the perfect squares 1 squared is 1 2 squared is 4 3 squared is 9 4 squared is 16 5 squared is 25 6 squared is 36 7 squared is 49 8 squared is 64 9 squared is 81 10 squared is 100 11 squared is 121 12 squared is 144 we're going to go up to 20 13 squared is 169 14 squared is 196 15 squared - 25 16 squared of 256 17 squared is 289 18 squared is um that's 3 21 actually 324 skis me and 19 squared is 361 20 squared is I'm going to put it here 400 so let's say if you want to simplify the square root of 8 which of these perfect squares which is the largest perfect square that goes into 8 its 4 8 divided by 4 is 2 so what you want to do is simplify root 8 as root 4 times root 2 and then you can take the square root of 4 which is 2 so your answer is 2 root 2 so let's try another example let's say if you want to simplify the square root of 18 which number goes into 18 what's the highest perfect square that goes into 18 it's none other than 9 18 divided by 9 is 2 so the square root of 18 is the square root of 9 times the square root of 2 and the square root of 9 we know it's dream so this simplifies to 3 root 2 now let's say if you want to simplify the square root of 50 25 times 2 is 50 and 25 is a perfect square so this simplifies to 5 root 2 so for the sake of practice go ahead and try these examples try the square root of 27 the square root of 48 the square root of 128 the square root of 450 and the square root of let's see 98 and of 1200 so feel free to pause the video and try these problems and then unpause it when you're ready so let's start with the first one what perfect square goes into 27 we know 9 goes into 27 nine times three is 27 and the square root of nine is three so this simplifies to 3 root 3 now what's the highest perfect square that goes into 48 4 goes into 48 but the same is true for 16 so if you start with the highest perfect square it's going to make simplifying it a lot easier especially when you're done with larger numbers 48 divided by 16 is 3 so the square root of 16 is 4 so this simplifies to 4 root 3 what perfect square goes into 128 128 is 64 times 2 then the square root of 64 is 8 now what about 450 this is 225 times 2 and the square root of 225 is 15 and for root 49 I mean root 98 that's 49 times 2 and the square root of 49 is 7 now what about 1200 what's the largest perfect square that goes into 1200 this would be 400 the square root of 400 well 3 times 400 1,200 and a square root of 400 is 20 see here 20 root 3 so that's how you can simplify perfect squares or radicals that have let's say an even index number if you don't see a number here this is invisible too now what about simplifying cube roots so you need to know your perfect cubes we're going to go over the perfect cubes at least up to 10 so 1 to the 3rd is 1 2 to the 3rd is 8 2 times 2 times 2 is 8 3 cube or 3 times 3 times 3 is 27 4 to the third power is 64 5 cube is 125 6 cube is 216 7q is 343 8 cubed is 512 9 cube is 729 and 10 to the third is a thousand ten times 10 times 10 is 1,000 so let's say if you want to find the cube root of 16 what is the highest perfect cube that goes into 16 well it's none other than 8 16 divided by 8 is 2 so you want to rewrite cube root of sixteen hours' the cube root of 8 times the cube root of 2 and the cube root of 8 is 2 because 2 to the third power is 8 so this simplifies to this answer so go ahead and try these examples the cube root of 50 for the cube root of 192 and the cube root of 375 so 27 goes into 54 and 27 times 2 is 54 and the cube root of 27 is 3 so this is 3 cube root of 2 now what number goes into 192 you can take each perfect cube and see which one is the visible body and 192 and pick the largest of them 8 goes into 192 and 64 but we're going to choose 64 I'm 64 92 divided by 64 is 3 so when you split it it's going to be 64 times 3 and the cube root of 64 is 4 so we get 4 cube root of 2 here we have 375 because it's a 5 minute chances are it's divisible by 125 so 375 divided by 125 is 3 and the cube root of 125 is 5 okay this is what supposed to be a 3 so that answer was supposed to be 4 cube root of 3 I don't know why I put 2 maybe I got in the habit of writing shoes try these two the cube root of a thousand 24 and the cube root of 5,000 so the largest perfect cube that goes into a thousand 24 is 512 512 is half of 1024 so we can rewrite this as 512 times 2 and the cube root of 512 is 8 so this simplifies to 8 cube root of 2 for this one 5,000 is a thousand times 5 and we know the cube root of a thousand that's 10 so we get 10 cube root of 5 now the next thing need to know is them the fourth powers let's say one to the fourth is 1 2 to the fourth is 16 3 to the fourth is 81 4 to the fourth power is 256 5 to the 4th power is 625 and 6 to the fourth power is 1296 so let's say if we have these three problems the 4th root of 32 and a fourth root of 162 and the 4 fruit of 768 so feel free to pause the video and try these practice problems so which of these numbers go into 32 clearly has to be 16 16 times 2 is 32 and we know the fourth root of 16 is 2 so we get 2 fourth root of 2 and 81 goes into 162 so we have the fourth root of 81 times the fourth root of 2 and a fourth root of 81 is 3 so we get 3 times the fourth root of 2 now which number goes into 768 let's see 768 divided by 81 is the decimal answer so that doesn't work 768 divided by 256 is 3 so we can rewrite this as the 4th root of 256 times the 4th root of 3 the four fruit of c-56 is four and that's the answer we get for that problem now let's add some variables to the to the mix let's say if we want to simplify the square root of 50 well before we use numbers let's just keep it simple how would you simplify the square root of x to the third now keep in mind there's a two here one way you can do this is you can split it to the square root of x squared times the square root of X the square root of x squared is just x times root X now depending on the level of depending on the algebra course that you're taken some teachers will want you to put an absolute value if if you have an even on like index number and if you get an odd exponent some teachers may want you to put it in absolute value other teachers they may not focus on that concept but depending on the course you take I'm going to add the absolute value just so you know when to do so so let's say if you have the square root of x to the stuff here's another way you can do it how many times does two going to 7 2 goes into 7 three times so it's going to be X cube and then with one remaining X to the one another way to show your work is you can separate it as X to the sixth times the square root of x here this is a 2 6 divided by 2 you can rewrite that as X 6 over 2 which is X to the 3rd times root X but I'm going to stick to one method so let's say if you want to find the square root of x to the 9 how many times is to going to 9 2 goes into 9 4 times with 1 remaining so let's say if you have the cube root of x to the 13 how many times does 3 go into 13 3 goes into 13 4 times with 1 remaining but this time we get up to 3 let's say if you have the cube root of x to the 11 how many times does 3 go into 11 3 goes into 11 3 times because 3 times 9 I mean 3 times 3 is 9 and the difference between 11 and 9 is 2 so 3 goes into 11 3 times with 2 remaining so using what you know go ahead and try these practice problems find the square root of x to the 5th the cube root of x to the 17th the 4th root of x to the 15th and the 7th root of x to the 19th so let's begin how many times is 2 going to 5 2 goes into 5 2 times because 2 times 2 is 4 and 5 minus 4 is 1 so the remainder is 1 now how many times does 3 go into 17 3 goes into 17 5 times 3 times 5 is 15 3 times 6 is 18 which is too much so it goes into 17 5 times and 17 minus 15 is 2 so we have 2 remaining but we got to write the 3 here as the index number how many times does 4 go into 15 4 goes into 15 3 times 4 times 3 is 12 and 15 minus 12 is 3 so the remainder is also 3 now how many times does 7 go into 19 seven goes into 19 2 times because 2 times 7 is 14 and 19 minus 14 is 5 do you see the pattern that's the quick way to get the answer but remember if you want to show you work let's focus on this one what you could have done is you can rewrite it as square root of x to the fourth times the square root of x + 4 divided by 2 is 2 so you get x squared root X now for this example what I would have done if you want to show your work I would split it to 15 and to the cube root of x to the 15 times the cube root of x squared 15 plus 2 is 17 um but 3 goes into 15 five times evenly because 15 divided by 3 is 5 so that's how this turns out to be an X to the fifth cube root x squared for this one I would break it into the fourth root of x - 12 times the fourth root of x to the 3 12 plus 3 is 15 but 4 goes into 12 evenly 12 divided by 4 is 3 so then you'll get this answer and for the last one I would rewrite it as the 7th root of X to the 14th times the 7th root of x to the 5th 14 divided by 7 is 2 so you get x squared times the 7th root of x to the 5th so now now that you understand how to simplify radicals when there's a variable inside let's put everything together let's say if you want to find the square root of 50 x cubed y to the 5th feel free to pause the video and go ahead and simplify so there's a 2 here so I'm going to use the technique where we're going to separate the radicals and then simplify a perfect square that goes into 50 is 25 so 50 I'm going to write it as square root of 25 times square root 2 and X cube I'm going to write it as the square root of x squared and the square root of X Y to the fifth I'm going to write it as the square root of Y to the fourth times the square root of one so there's invisible two here so I had to separate the even values from these odd exponents the square root of 25 is 5 and the square root of x squared is X and the square root of Y to the fourth is y squared now if we need to add parentheses I probably going to add it for some other examples but if you want to add it here you would add it to this one because it has an odd exponent and then everything else that can't be simplified collect it and put it inside a single radical so the square root of 2xy that's the answer for of that example so let's try another one so let's say if you want to find the square root of 72 X to the fourth Y to the fifth Z to the sixth a perfect square that goes into 72 is 36 so we're going to write this as the square root of 36 times the square root of 2 so 2 goes to the 4 evenly so I'm just going to rewrite it as the square root of x to the 4th for y to the fifth it's an odd number so I'm going to write it as Y to the 4th times y to the first it adds up to 5 and then Z to the 6 I'm going to leave it alone the square root of 36 is 6 the square root of x to the fourth is x squared because the exponent is even and not on we don't need a absolute value the square root of Y to the fourth is y squared and the square root of Z to the 6 is Z to the third so we have an even index and out of these oh so we're going to put this in a pan absolute value then every other term that wasn't simplified in this case the 2 and the Y we're going to put it inside the the radical so 2 times y so that's the answer for that particular problem so now let's try a cube root so let's say if you want to simplify the cube root of 24 X to the sixth Y to the seventh Z to the 10r to the 13 so what perfect cube goes into 24 eight so 24 we're going to be write it as the cube root of eight times the cube root of three and three goes into six evenly so we're just going to keep like that cube root of x to the six seven doesn't go into six evenly but six does so we're going to write it as the cube root of Y to the 6 times the cube root of Y to the first because 6 plus 1 is 7 3 doesn't go into ten evenly but 3 goes into 9 so I'm going to be write it as the cube root of Z to the 9 times the cube root of Z to the 1 and times let me add it here 3 goes into 12 not 13 so cube root of our 12 times the cube root of our to the first one plus 12 is 13 okay let's simplify the radicals that can be simplified the cube root of 8 is 2 and here 6 divided by 3 is 2 so that turns out to be x squared if you have an odd eat index there's no absolute values that you have to write for this now this one 6 divided by 3 that's also Y squared we could simplify this one 9 divided by 3 is 3 and 12 divided by 3 is 4 so we get R to the fourth everything else is going to stay in the cube root so we have a 3 we have a Y we have a Z and we have our and so this is the answer for that problem so now let's try 2 for fruit let's say if we want to find a for fruit of 32 X to the 7 Y to the 12 Z to the 15 R to the 20 so what number goes into 32 that's a 4 fruit in this case 16 so we're going to rewrite this as the four fruit of 16 and times the fourth root of 2 now four days ago into 7 4 goes into 4 & 4 plus 3 is 7 so we're going to write it as X to the fourth times X to the third 4 goes into 12 so we're just going to rewrite it as as that 4 does it go into 13 it goes the highest number that it goes closest to 15 is 12 so we're going to write as Z to the 12 times the fourth root of Z to the third because 12 + 3 is 5 now for does go into 20 so we're just going to leave this alone so now let's simplify the fourth root of 16 what number 4 times what number times itself 4 times is 16 2 times 2 times 2 times 2 is 16 so the fourth root of 16 is 2 next we can simplify this 4 divided by 4 is 1 so we have an even exponent I mean an even index number and an odd exponent so we're going to put that in parenthesis in absolute value the fourth root of Y to the 12 is y to the third and the fourth root of Z to 12 is Z to the third and the fourth root of R to the 20 is R to the fifth so we can put this whole thing in an absolute value so everything else will just come and collect it now inside one radical so we have a 2 we have X cubed and we have Z to the third and that's our answer so now let's say if you get a problem that looks like this eight divided by a radical three and you want to simplify it what can you do in a situation like this all you can do is rationalize the denominator so what you want to do is multiply top and bottom by the square root of three so you're going to get 8 root 3 divided by root 9 3 times 3 is 9 but the square root of 9 is 3 so this is the answer that you get so let's say if you have 5 square root X what would you do multiply the top and bottom by root X so you're going to get 5 root X times X and that's it for them so let's say if you have seven divided by the cube root of four so this is like 4 to the first power what would you do in such a situation now notice this is what you can't do you don't want to multiply top and bottom by the cube root of 4 the reason being is on the bottom you'll get the cube root of 16 and you can't really get rid of the radical because 16 is not a perfect cube you could simplify it but you want to get rid of the radical so instead you want to do something different you see notice that there's a 3 here but you only have one for inside you need two fours to make to have a total of three fours to they get it out of the radical so what you want to do is multiply top and bottom by the cube root of 4 squared because 4 to the first power times 4 squared is 4 cube which is 64 and the cube root of 64 is 4 so right now we have 7 times the cube root of I guess in this point 4 squared we can write it as 16 and on the bottom we have the cube root of 4 to the third we can make that 64 but we don't need to the threes will cancel and the radical will disappear so we'll just get 4 in the bottom now the cube root of 16 we can simplify that because 8 is a perfect cube so we can break it down as the cube root of 8 times the cube root of 2 and the cube root of 8 is 2 so we can replace it with times 2 cube root of 2 and 2 divided by 4 we can simplify if we divide it backwards 4 divided by 2 is 2 so we'll put that 2 on the bottom and so this is what you get that's the simplified answer for that problem so let's say if you have nine divided by the fourth root of x to the first power so what you want to do is multiply top and bottom by the fourth root of x cubed because one plus three is four so you're going to get nine for fruit of X cubed divided by the fourth root of one plus three is four so X to the four so these will cancel and then you'll get nine for fruit of X to the third over X so that's how you can rationalize that denominator so now let's say if we have this problem five divided by the seventh root of x squared we need to get up to seven so two plus five to seven so we're going to multiply by the seventh root of x to the fifth top and bottom so this is going to be a final answer well not yet dough but initially we're gonna get the seventh root of x to the 7 when those two numbers are the same the radical disappears that's our answer try this one let's say if you have three divided by the ninth root of x squared Y to the fourth Z to the seventh how would you rationalize the denominator in such a situation so what you want to do is multiply top and bottom by the ninth root of something now you want all of the exponents to add up to 9 9 minus 2 is 7 so you need X to the 7 9 minus 4 is 5 so you need Y to the 5 9 minus 7 is 2 so these are the values that you need so you get 3 times the ninth root of x to the 7 Y to the fifth Z squared divided by the ninth fruit of X to the 9 Y to the 9 z to 9 so all the nines will cancel so this is 3 9 fruit of X to the 7 for y to the 5th Z squared over X Y Z so now you know how to rationalize denominators under any situation or circumstance that you might find yourself in what about this problem you let's see if you get this problem like on a test or something or in your homework assignment what would you do to simplify it if you see a radical separated by like it is a number and a radical separated by a plus or minus sign multiply top and bottom by the conjugate of the denominator so since we have four and minus root three the conjugate is four plus root 3 change the minus to a plus on the top we're going to distribute the 15 15 times four is 60 and 15 times root 3 is just 15 root 3 on the bottom we're going to foil 4 times 4 is 16 4 times root 3 is 4 root 3 and 4 times negative 3 is negative 4 root 3 and negative root 3 times root 3 is like negative square root 9 which is just negative 3 because initially you get this 3 times 3 is 9 and then the square root of 9 is 3 so you get negative 3 so notice that the two middle terms will always cancel every time you multiply by the conjugate so what we now have is 60 plus 15 root 3 divided by 16 minus 3 which is 13 so that's how you simplify it so for the sake of practice try this 110 divided by root 5 plus root 2 so what we need to do is multiply top and bottom by the radical I mean the conjugate of the bottom so on the top we're going to distribute the 10 so it's going to be a 10 root 5 minus 10 root 2 on the bottom we're going to foil root 5 times root 5 is just 5 now we know the two middle terms will cancel so we'll just multiply the last two so root 2 times root 2 is just 2 it might be better to keep in this factored form while we're at it like this and then 5 minus 2 is 3 so that's our answer okay try this one 3 minus root 2 divided by 5 plus root 2 so once again let's multiply top and bottom by the conjugate knot of the numerator but of the denominator because we want to get rid of all radicals in the denominator so if we fall you the numerator 3 times 5 is 15 and 3 times radical 2 is negative 3 radical 2 here 5 times root 2 is negative 5 root 2 and the two negative signs will cancel root 2 times root 2 is just 2 on the bottom notice that they're conjugates of each other so to the two middle terms will cancel so 5 times 5 is 25 and root 2 times negative root 2 is just minus 2 so let's combine like terms 15 plus 2 is 17 and negative 3 minus 5 is negative 8 review 2 and on the bottom what we have is 23 so that's our answer for that problem so next we're going to go over add and and subtracting radicals so let's say if you have a problem that looks like this 3 square root of 18 minus 4 root 50 minus 5 root 32 how would you simplify so first let's break down the radicals into perfect squares a perfect square that goes into 18 is nine so radical 18 I'm going to be write it as radical nine times radical two a perfect square that goes into 50 is 25 so that's 25 and 2 and the perfect square that goes into 32 is 16 and 32 divided by 16 is 2 the square root of 9 is 3 and the square root of 25 is 5 and the square root of 16 is 4 so 3 times 3 is 9 square root 2 and 4 times 5 is 20 times the square root of 2 and 5 times 4 is also 20 square root 2 so at this point we can add the exponents this situation is similar to let's say if you have like 9x + 5 X you add 9 + 5 you get 14 and you keep the variable the same imagine the square root as an X when you add the coefficients the square root is going to remain the same so 9 minus 20 is negative 11 and negative 11 minus 20 is 19:31 so it's negative 31 square root 2 let's try another example like this but one involve in cube roots so let's say if you have 12 times the cube root of 16 minus 2 times the cube root of 50 4 plus 5 times the cube root of 128 go ahead and try this problem and feel free to simplify it when you're ready so what perfect cube goes into 16 8 is a perfect cube so we're going to rewrite it as cube root of 8 times the cube root of 2 and for the other one 27 is a perfect cube that goes into 54 and 54 divided by 27 is 2 and 64 goes into 128 so this is what we now have so what is the cube root of 8 what times what times what is 8 - 2 times 2 times 2 is 8 now what about the cube root of 27 what times what times what is 27 the cube root of 27 is 3 3 to the third power is 27 so this is why you need to know your perfect squares your perfect cubes and some of the the 4 power ones the cube root of 64 is 4 and so now we can multiply and then simplify 12 times 2 is 24 so we have 24 cube root of 2 2 times 3 of 6 and 5 times 4 is 20 so now what we're going to do is combine the coefficients 24 minus 6 is 18 and 18 plus 20 is 38 so we have 38 cube root of 2 that's the answer so now you know how to add and subtract radicals so next we're going to go over like how to multiply radicals so let's say if you have the square root of 5 times the square root of 6 this is just the square root of 30 and we can't reduce the square root of 30 because there's no perfect square except one that goes with the square root of 30 so that's all we can do for that problem but now let's say if you have this one the square root of 12 times the square root of 32 how would you do this problem without a calculator would you multiply 12 and 32 now you could do that and you might get a huge number that could be like 300 or 400 and you're gonna have to simplify that large number or you can make your life easier and simplify it before you multiply what perfect square goes into 12 4 so break down the square root of 12 as 4 times 3 and what perfect square goes into 32 6 I mean not 6 but 16 32 is 16 times 2 and then now you want to simplify the square root of 4 is 2 and the square root of 16 is 4 so now what you want to do is multiply the 2 and the 4 the numbers on the outside that's going to give you 8 times the numbers that are inside the radical 2 times 3 is 6 and this is your answer it's a much easier if you simplify first before you multiply so let's try another example let's say if you want to simplify or multiply 5 root 20 by 7 root 18 so don't multiply 20 by 18 yet simplified first so a perfect square that goes into 20 is 4 so 4 times 5 is 20 and the perfect square that goes into 18 is 9 and 2 well 9 is the perfect square but 18 divided by 9 is 2 so now let's simplify the radicals that we can simplify the square root of 4 is 2 and the square root of 9 is 3 so now let's multiply these 2 numbers and those 2 5 times 2 is 10 and 7 times 3 is 21 now what's 10 times 21 10 times 21 is 210 you just got to add a 0 and 5 and 2 is 10 so we get 2 10 times the square root of 10 so what if you get a problem that looks like this the square root of three over five times the square root of five over seven how would you simplify it so you can just multiply across we can't really simplify three and five they're not perfect squares so we can combine this into a single radical what we have is 3 times 5 divided by 5 times 7 we can cancel the 5 5 divided by 5 is 1 so our final answer is just well right now we have the square root of 3 over 7 which is the same as root 3 divided by root 7 now most teachers they want you to rationalize it so we're going to multiply the top and bottom by the square root of 7 so we're going to get root 21 over 7 and that's our answer so let's continue so let's say if you have this problem the square root of 8 divided by 27 times the square root of 30 divided by 12 so instead of multiplying by 8 and 30 let's simplify let's see what we can cancel so 30 on top we have 8 times 30 but 30 you can break it down into 10 and 3 and 10 is 5 times 2 and then times 3 so 5 times 2 is 10 10 times 3 is 30 so 30 is 5 times 2 times 3 8 we can break that down into 4 times 2 27 we can break that down 2 3 times 3 times 3 that's 27 and 12 we can break it into 4 times 3 so you want to simplify first so we can cancel a 4 and we can cancel one of the threes so what we're left with is the square root of 2 times 5 times 2 which is like just the square root of 20 and on the bottom we still have the square root of 27 so perfect square that goes into 20 is 4 so we have square root 4 times the square root of 5 because 20 divided by 4 is 5 as we can see here these twos they make the 4 and here's the remaining 5 and a perfect square that goes into 27 is 9 27 divided by 9 is 3 so the square root of 4 is 2 so we have 2 root 5 on top and the square root of 9 is 3 so we have 3 root 3 we have to rationalize it so we're going to multiply the top and bottom by root 3 so on the top we have 2 times root 15 5 times 3 is 15 and 3 times the square root 3 times the square root of 3 is square root of 9 which becomes to me so our final answer is 2 15 divided by 9 so now let's say if you have a problem that just looks like this the square root of 25 divided by the square root of 36 what's the answer for this one simplify it one step at a time the square root of 25 is 5 and the square root of 36 is 6 and that's all you got to do for that one now what about this one the square root of 200 divided by 12 so before we we try to like simplify the radical let's divide first we could break down we can divide both numbers by 2 let's simplify the fraction 200 divided by 2 is 100 and 12 divided by 2 is 6 now notice that we can take the square root of 100 the square root of 100 is 10 and then we can multiply top and bottom by root 6 so you get 10 root 6 over 6 and you can reduce that further you can divide 10 by 2 and 6 by 2 so you get 5 root 6 divided by 3 and that's the answer for that one now what about this example the square root of 40 over 55 notice that 40 and 55 are divisible by 5 so let's divide by 5 first so we get the square root of 8 over the square root of 11 now radical 8 we can write that as square root of 4 times square root of 2 because 4 is a perfect square and on the bottom we have the square root of 11 so the square root of 4 is just 2 and then we got to rationalize it so let's multiply top and bottom by root 11 so 2 times 11 is 22 and this is our answer try this example the square root of 18 X to the 7 y squared divided by 48 X to the third Y to the fifth so there's many ways we can do this we could divide first or we could simplify the radical first what would you do for this example let's divide first let's divide the coefficients by two so we're going to get 18 divided by 2 is 9 48 divided by 2 is 24 and now let's simplify the exponent so when you divide exponents like X to the 7 divided by X to the third you have to subtract 7 minus 3 is 4 so we get X to the fourth on top now y squared divided by Y to the fifth that's 2 minus 5 which is negative 3 on the top but you can move it back to the bottom but as you do so you change the negative exponents will a positive exponent so we have Y to the 3rd on the bottom and we still have the square root now those 2 smaller square roots is the same as if I wrote largest the square root so you can write it either way the square root of 9 is 3 and the square root of x to the 4 4 divided by 2 is 2 so you just get x squared now for the bottom part 24 we can break that down into 4 times 6 because 4 is the perfect square and y cube we can break that into square root y squared times the square root of Y the square root of 4 is 2 and the square root of Y squared is just Y and then what's left over is the 6 and the Y which is just root 6 1 so now we're going to rationalize it we're going to multiply top and bottom by a root 6 1 so on top we're going to get 3 x squared root 6 Y on the bottom we have 2y times 6 1 so our final answer is three x squared root six Y over twelve Y squared actually we could be do step twelve divided by three if you divide it backwards you get four so there's going to be a four and a bottom for y squared and on top x squared root six one okay now that's the final answer so our final example for this video is going to be this problem the square root of 75 times X to the eighth Y cube over 108 x squared Y to the seventh so let's simplify the radical first before dividing this time let's do it differently 75 we can break that out into 25 and 3 2 divided by 8 or I mean 8 divided by 2 is in this case X to the fourth and 2 goes into 3 one time so we got Y to the first and there's one remaining so that's going to stay inside the radical now 108 36 goes is 108 108 divided by 36 is 3 so we can rewrite the square root of 108 times the square root of 36 I mean square root of 36 times the square root of 3 2 goes into 2 one time so we just get X to the first 2 goes into 7 three times with one remaining so these cancel and the 3s cancel so they just disappear the square root of 25 is 5 and the square root of 36 is 6 now X to the fourth divided by X to the first you subtract 4 minus 1 you get X to the third and here if you subtracted backwards 3 minus 1 is 2 but that has to go to bottom that's another technique of simplifying those types of problems so this is our final answer 5x cubed divided by 6 y squared so that's really it for this video I mean we covered everything or at least most topics leave actually there's only one other thing I want to mention let's say if you have the square root of a negative number now this is not a real solution this is an imaginary solution the square you can write this as a square root of 4 times the square root of negative 1 the square root of 4 is 2 and the square root of negative 1 is I so let's say if you have the square root of negative 8 a perfect square that goes into 8 is 4 so you have 4 times 2 times the square root of negative 1 square root of 4 is 2 and the square root of negative 1 is I see get that answer so now I believe we've covered every topic relating to radicals you know how to add and subtract radicals you know how to multiply and divide radicals you know how to simplify United rationalize and if you see any negative numbers inside you don't get a real solution but you could write it as an imaginary solution just keep in mind I is the square root of negative 1 I squared is negative 1 I cube is negative I and I to the 4th is 1 if that becomes relevant sometime in the future there's actually another example that I almost forgot to go over but it is pretty relevant to this video so let's say if you have the cube root of x to the 7 and you wish to multiply it by the fifth root of x to the third how would you do it now let's say if you have the square root of 5 and if you want to multiply it by the square root of 6 5 times 6 is 30 and it seems pretty straightforward but the only reason why we can do that is because the index number is the same so let's say if we want to multiply the square root of x cubed times X to the fifth X cubed times X to the fifth is X to the 8th you just got to add the exponents and that reduces to X to the fourth only because the index numbers are the same are we allowed to multiply by what's inside but in this scenario the index numbers are different so we can't just say x times 7 I mean X to the 7 times X cubed and right X to the 10 because well index number would you write 3 for 5 or something else so then how do we simplify a problem like this when the index numbers are different what do we do what you need to do is convert it from its radical form to its exponential form the cube root of x to the 7 is the same as X to the 7/3 and the fifth root of x cubed is the same as X raised to the 3 over 5 this number becomes the numerator and this number becomes the denominator now when you multiply two variables let's say like x squared times X cube what you have to do is add the exponents two plus three is five so here we have the same base which is X so to multiply these two variables we have to add seven over three plus three over five and the only way to do that is to get common denominators so I'm going to multiply this side by 3 over 3 and the other fraction by 5 over 5 so what we now have is a 35 over 15 plus 9 over 15 so this is X 35 over 15 times X 9 of 15 now that we have the same denominator we can add the numerators 9 plus 35 is 44 so this is X raised to the 44 over 15 now you want to simplify how many times does 15 go into 44 15 goes into 44 2 times 30 is fifths 30 is divisible by 15 so I'm going to write 44 and 15 as 30 over 15 times 14 over 15 because 30 plus 14 is 40 44 30 divided by 15 is 2 and the last one we can't simplify it so we can rewrite it as the 15th root of x to the 14th now another way you could solve this problem is at this point you can convert it back to radical form which is 15 the 15th root of x to the 44 and then simplify using the techniques that we've covered in this video how many times is 15 go into 44 it goes into it 2 times 15 times 2 is 30 but 15 times 3 is 45 which is which exceeds 44 so 44 minus 30 is 14 and you end up getting the same answer so let's try another example so let's say if you have the fifth root of x to the fourth and you wish to multiply it by the cube root of x to the eighth using what you know feel free to pause the video and give this problem a shot go ahead and try it so let's begin let's rewrite it as X to the 4 over 5 times X to the eighth over three and let's get common denominator so we're going to multiply this fraction by 3 over 3 and this one by 5 over 5 because the common denominator is 15 so we're going to get 12 over 15 times 40 over 15 and 12 plus 40 is 52 so we have X 52 over 15 which is the same as the 15th root of x to the 52 now how many times does 15 go into 52 15 goes into 52 at least 3 times because 15 times 3 is 45 15 times 4 is too much that's 60 so we're going to get X cubed and the remainder 452 minus 45 is 7 so there's 7 X variables remaining so this is our answer so let's say if we have a number instead of a variable let's say if we wish to multiply the cube root of 16 times the square root of 12 now what I would do is before you multiply I would simplify to make your life easier the cube root of 16 is the same as the cube root of 8 times the cube root of 2 and the cube root of 12 I mean the square root of 12 is the square root of 4 times the square root of 3 we could simplify these two because the cube root of 8 is a perfect cube that's 2 and the square root of 4 is also 2 so what we really need to combine is these the cube root of 2 times the square root of 3 so 2 times 2 is 4 now there is an invisible one that you don't see this is 3 to the 1 and 2 to the 1 and here if there's no index number into 2 so we now have is 2 to the 1/3 times 3 to the 1/2 so we can only multiply the bases if the exponents are the same so for example let's say if you have 4 times 5 is 20 the reason being is because the exponents are the same but you can't say 4 squared times 5 cube is 20 because what exponent would you have so that doesn't work but let's say if you have 4 squared times 5 squared that's equal to 20 squared because the exponents are identical so you can only multiply the bases the 4 and the 5 if the exponents are identical and you can only add the exponents if the bases are identical for example 4 squared times 4 cubed is 4 to the fifth so whenever you perform an operation something has to remain the same while you change the other thing either the base has to be the same or the exponents so right now the bases are different so to combine the 208 we got to make the exponent so we have to get similar exponents so let's see if that's possible actually that that's not possible there's no way we can get the same exponents for this problem let's say if you were to get common denominators this would be 2 over 6 and the other one would be 3 over 6 2 over 6 is the same as 1/3 3 over 6 is 1/2 so because the let's say if you have 2 & 2 over 6 times 3 & 3 over 6 neither the exponents or the bases are identical so we can't combine that so our final answer is just 4 cube root of 2 times the square root of 3 we can't combine those now let's say if we had this problem in the cube root of 2 times the square root of 2 this we can't combine because the twos are identical so 2 to the 1/3 times 2 to the 1/2 so the 1/3 I'm going to multiply by 2 over 2 and the 1/2 I'm going to multiply by 3 over 3 so this would be 2 raised to the 2 over 6 times 2 raised to the 1 the 3 over 6 so now we can add the exponents so this is going to be 2 raised to the 5 over 6 which is the 6 root of 2 to the 5 or 2 to the fifth power so only if the bases are the same can we multiply two radicals over different index so try this one the cube root of 4 times the fifth root of 4 so we have 4 to the 1/3 times 4 is + 1 here to the 1/5 so I'm going to multiply this one by 5 over 5 and this by 3 over 3 so we now have 4 raised to the 5/3 times 4 I mean not 5/3 about 4 and raised to the 5 over 15 times 4 raised to the 3 over 15 three plus five is eight so we get 4 to the 8th over 15 which is the 15th four to the eighth power but it turns out we can actually simplify further because 4 is equal to 2 squared so we have the 15 root of 2 squared raised to the 8th power now when you raise one exponent to one another exponent you need to multiply it so 2 times 8 is 16 so this becomes the 15th root of 2 to the 16 and 15 does go into 16 at least once so 1/2 comes up and a remainder is 1 so we get 2 times the 15th root of 2 that's the final answer for that problem now what about for dividing by 2 radicals with a different index so let's say if you have the 4th root of x to the 9 divided by the cube root of x squared how would you do this one so first let's convert it to exponential form this is X raised to the 9 over 4 and this is X raised to the 2/3 so now we need to get the common denominators so we're going to multiply this 1 by 3 over 3 and this by 4 over 4 the common denominator between 4 & 3 is 12 so this is going to be 3 times 9 is 27 3 times 4 is 12 and here we have 8 over 12 so now when you divide you got to subtract the exponents 27 minus 8 is 19 and the 12 is going to be the common denominator so that stays the same so now we can put that back into radical form so that's the 12th root of x to the 19th so how many times does 12 go into 19 12 goes into 19 one time and there's seven remaining so we still need the 12 through here so here we have an even index and odd exponent so it's the absolute value of x 12 root of x to the 7 now what about this one the cube root of 32 divided by the fourth root of 32 let's simplify this one so this is 32 to the 1/3 divided by 32 to the 1/4 so the common denominator is 12 we're going to multiply this by 4 over 4 and this 1 by 3 over 4 so on top we have 32 raised to the 4 over 12 and 32 raised to the 3 over 12 4 over 12 minus 3 over 12 is 1 over 12 so we get 32 to the 1/12 so that's basically the 12 root of 32 so that's the answer for that example so now you know how to simplify radicals especially if there's different index numbers you know how to multiply or divide radicals with different index numbers but by the way make sure you understand that you can't really add or subtract radicals with different index numbers like the cube root of 2 plus the 4th root of 2 you can't simplify if they're separated by addition or subtraction 2 to the 1/3 plus 2 to the 1/4 there's nothing you can really do here it's like saying X cubed plus X to the 4 if you can't add them you just leave them the way they are so you can only simplify radicals with different index numbers if you're dealing with multiplication or division not addition and subtraction so that Club is this video and thanks for watching and have a great day