in this lesson we're going to focus on dividing radicals let's divide 64 by x squared and let's simplify this example so how can we simplify this radical the square root of 64 is 8. the square root of x squared is simply x but absolute value of x since we have an even index number and we got an odd exponent as a result now what about this example the square root of 50x divided by the square root of 2x how can we divide and simplify this example so first we could divide 50 by two that's going to be 25 x divided by x will cancel that's one so what we have is the square root of 25 which is 5. what about this example what is the cube root of 27 a to the ninth divided by eight b to the sixth so first we could take the cube root of 27 which is three the cube root of eight is two the cube root of a to the nine basically you take the exponent and divide by three so it's a to the third power and the cube root of b to the sixth is b to the sixth over three which is two so the answer is three a cubed over two b squared what about this one the square root of 75 x to the eighth y to the tenth divided by 48 a to the fourth b to the fifth the square root of 75 we can break that into square root 3 times the square root of 25 the square root of x to the 8 we just need to divide 8 by 2 so it's x to the fourth and the square root of y to the tenth is y to the fifth but because we got an odd exponent from an even index number we need to put it using absolute value notation root 48 we can break that into square root 3 and square root 16. the square root of a to the fourth is a squared and the square root of b to the fifth that's basically two doesn't go into five evenly two goes into five two times with one remaining so therefore we're gonna have a b inside the radical now we can cancel the square root of three the square root of 25 is 5. so we have 5 x to the fourth y to the fifth and the square root of 16 is 4. now we need to get rid of the radical on the bottom so let's rationalize it let's multiply the top and the bottom by root b so the final answer is 5 x to the fourth y to the fifth square root b divided by now these two will become b and b times b squared is b cubed so it's four a squared b to the third power that's the final answer here's another example simplify the square root of 36 x to the 9th y to the 11th divided by 25 x cubed y to the fourth the square root of 36 is 6 and the square root of 25 is 5. now what about x to the 9 divided by x cubed when you divide exponents or rather when you divide by a common base you need to subtract the exponents 9 minus 3 is 6. so this becomes the square root of x to the sixth which is x to the third and we need to put it using absolute value now y to the 11th divided by y to the fourth that's basically y to the seventh and this is y to the sixth times y and the square root of y to the sixth is y cubed so on top we have the absolute value of y cubed times root y so we can write the final answer as six absolute value x cubed y cubed times root y divided by five try one more example the cube root of 16 a to the ninth b to the negative six c to the fifth over 54 a to the negative two b to the seven and c to negative eight sixteen is basically two times eight and fifty-four is two times twenty-seven now eight to the nine divided by a to the negative two we need to subtract the exponents nine minus negative two is positive eleven so it's going to be eight to the eleventh on top b to the negative 6 divided by b to the 7th we need to subtract negative 6 by 7 and that's negative 13. so this is b to the negative 13 on top but if we move it to the bottom the negative exponent will become positive so it's b to the 13 on the bottom and then we have 5 minus negative 8 which is positive 13. so it's c to the 13 on top now we can cancel two and the cube root of eight is two the cube root of 27 is three now what is the cube root of a to the eleventh this we can break it up into nine and two the cube root of nine is three nine divided by three is three so we get a cubed cube root a squared now what about the cube root of c to the thirteen three goes into thirteen four times and four times three is twelve thirteen minus twelve is one so we have one remaining so on top is going to be c to the fourth cube root c and the cube root of b to the 13 is going to be just like the cube root of c to the 13. it's going to be b to the fourth cube root times b now let's see if we can simplify all of this so we have two a to the third c to the fourth and then we can combine these two so that's going to be the cube root of a squared times c divided by three b to the fourth cube root of b the last thing that we need to do is rationalize the denominator let's multiply the top and the bottom by the cube root of b squared on the top we'll have two a to the third c to the fourth cube root a squared b squared c and on the bottom we're going to have three b to the fourth and then let's focus on this part b times b squared is b to the third and the cube root of b to the third will become b so this is going to be b to the fourth times b which we can combine it as b to the fifth so therefore this is going to be the final answer 2 a cubed c to the fourth times the cube root of a squared b squared c divided by three b to the fifth now let's work on one more example let's say if we have the fourth root of 81 x to the 12 y to the 11 z to the negative 3 divided by 32 x to the fourth y to the negative nine z to negative seven go ahead and simplify this problem so the fourth root of 81 is three now what about the fourth root of 32 well 32 can be broken up into five twos it's basically two to the fifth power and if we take the fourth root of two to the fifth power we can take four twos and take it out as one so it's going to be two times the fourth root of two so we're gonna have a two on the outside and one is still will be left over on the inside so i'm going to write it like this for now now i'm going to divide everything on the inside first so we still have a 2 on the inside at the bottom now x to the 12 divided by x to the four 12 minus four is eight so this is going to be x to the eight and then eleven minus negative nine that's the same as eleven plus nine so that's gonna be 20. so y to the 11 divided by y to the negative 9 is y to the 20th power and then negative 3 minus negative 7 that's negative 3 plus 7 which is plus four so this part is going to give us z to the fourth power now the fourth root of x to the eight is going to be x squared what you need to do is divide eight by 4 you get 2. so x squared is going to come on the outside 20 divided by 4 is 5. so y to the fifth is going to come out and then 4 divided by 4 is 1 so z to the first power will come out and we can put this within an absolute value symbol because we have an even index number and an odd exponent now right now we have the fourth root of one over two so we need to rationalize this so to do that i'm going to multiply the bottom and the top by the fourth root of two to the third power two to the third power is eight and on the bottom we're gonna have two times eight which is sixteen and the fourth root of sixteen is two so this becomes two or you can see it as one plus three is four and then if you have the fourth root of 2 to the 4 then 4 divided by 4 is 1 so you get 2 to the 1 which is the same as 2. so we can replace the 4th root of one over two with the fourth root of eight divided by two so the final answer is going to be three x squared absolute value y to the fifth times z times the fourth root of eight all divided by two times two which is four and so that's it for this video now you know how to divide rational expressions and simplify them as well you