in this video we're going to focus on how to simplify radicals with variables and exponents so let's say if you want to simplify the square root of x to the fifth the index number is a two now one way you can do this is you can write x five times and because there's a two you need to take out two at a time so this will come out as one x and this will come out as another x and you're gonna get x times x square root of 1 just x by itself so this is equal to x squared root x now another way you can simplify this or get the same answer is if you do it this way how many times does two go into five two goes into five two times because two times two is four two times three is six that's too much and what's remaining five minus four is one so you get one remaining and that's another way you can simplify it so let's say for example if you want to simplify the square root of x to the seven how many times does two go into seven two goes into seven three times with one remaining now let's try this one how many times does two go into eight two goes into eight or eight divided by two is four two goes into eight four times with no remainder two goes into nine four times and 2 goes into 12 6 times with no remainder for the 9 there's a remainder of 1 so the y is still on the inside that's a quicker way that you can use to simplify radicals now let's say if you have a number let's say if you want to simplify the square root of let's say um 32 what you want to do is break this down into two numbers one of which was is a perfect square so 32 can be broken down into 16 and 2. now the reason why i chose 16 and 2 is because we know what the square root of 16 is and that's 4. and so this is just 4 root 2. now let's say if we have a problem that looks like this let's say if we want to simplify the square root of 50 x cubed y to the 18th so how many times does two go into three two goes into three one time with uh one remaining and two goes into eighteen nine times now usually when you have an even index and an odd exponent you got to put it in absolute value now your teacher may not go over this but some teachers do but just in case if you have one of those teachers who wants you to use an absolute value you only need it if you have an even index and if you get an odd exponent after it comes out of the radical now the only thing we have to simplify is root 50. square root 50 we can break it down into square root 25 and 2 because 25 times 2 is 50. and the square root of 25 is 5 but the 2 stays inside the radical so we can put a 5 on the outside and let's put the 2 inside so this is the final answer that's how you can simplify that expression let's try some other problems so let's say if we have the cube root of x to the fifth y to the knife and z to the fourteenth so how many times does three go into five three goes into five one time with uh two remaining so we're gonna put x squared inside and the index number would it's going to stay 3. now how many times is 3 going to 9 9 divided by 3 is 3 with no remainder and how many times does 3 go into 14 3 goes into 14 four times and 3 times 4 is 12 so 14 minus 12 is 2 so we have 2 remaining so that's how you can simplify radicals let's try one final problem feel free to pause the video and see if you can see if you can get the answer for this one so the cube root of 16 x to the 14th y to the 15th z to the 20th so how many times does 3 go into 14 3 goes into 14 4 times 3 times 4 is 12 and 14 minus 12 is 2 so we're going to get x squared on the inside 3 goes into 15 five times with no remainder because 15 divided by 3 is 5. 3 goes into 20 six times 3 times 6 is 18. three doesn't go into twenty evenly and twenty minus eighteen is two so there's two remaining now let's simplify the cube root of sixteen perfect cubes are one one cube is one two to the third power is eight three to the third power is twenty seven so a perfect cube that goes into sixteen is eight so sixteen divided by eight is two so you want to write cube root of 16 as the cube root of 8 times cube root of 2 because the cube root of 8 simplifies to 2. so this 2 is going to go on the outside which we're going to put it here and this 2 remains on the inside which i'm going to put it there so this is our final answer for that problem so that's how you can simplify radicals with variables and exponents but actually let's try one more let's say if you have a question it looks like this let's say the square root of 75 x to the seventh y to the third z to the tenth over eight let's say x to the third y to the ninth z to the fourth so the first thing we can do is um let's simplify everything let's rewrite it so 75 is uh 25 times three we can square root 25 that's five but we'll do that later and eight is four times two because we can take the square root of four now when you divide exponents i mean when you divide variables you gotta subtract the exponents seven minus three is four and that goes on top because there's more x values on top now for this one you can do three minus nine but i think it's easy if you subtract it backwards the nine minus three which is six and because we subtract it backwards the six goes on the bottom and then ten minus four so there's more z's on top down the bottom so we're gonna put it on top so z to the sixth and now let's simplify it the square root of 25 is five and the square root of four is two now two goes into four two times four divided by two is two so we get x squared two goes into six three times so we get z to the third and six divided by two is three so we get y to the third and inside the radical we still have a radical three and the square root 2 left over so now we also need to add some absolute values because we have an even index and we have a few odd exponents we need to put z an absolute value and a y so our last step is to multiply top and bottom by square root of two we need to rationalize the denominator we need to get rid of that radical so our answer our final answer is five x squared absolute value of z to the third square root six over square root two times square root two is is square root of four which simplifies to two and two times two gives us four so we get four absolute value y cubed this is our final answer for that particular problem okay let's try just one more problem so let's say if we have the cube root of 16 x to the seven y to the fourth z to the ninth divided by 54 x squared y to the knife z to the 15th so feel free to pause the video and try this example yourself so the first thing i would do is within a radical i would divide both numbers by two referring to the sixteen and the fifty four so right now what i have is the cube root of eight which is a perfect cube over 27. 16 divided by 2 is 8 half of 54 is 27. so now what i'm going to do is subtract the exponents 7 minus 2 is 5. and for the y's i'm going to subtract it backwards 9 minus 4 is 5. so y to the fifth and for z i'm going to subtract it backwards 15 minus 9 is 6 but that's going to go on the bottom and so now we can simplify it so the cube root of 8 is 2 and 3 goes into 5 only one time with two remaining and the cube root of 27 is three and three goes into five one time just like x with two remaining three goes into six two times so that becomes z squared now we don't need any absolute values because this is an odd index we only need it for even index numbers that produce an odd exponent so now let's simplify what we have so we need to get rid of the radical on the bottom so we're going to multiply top and bottom by the cube root of y to the first power so what we now have is two x cube root x squared times y divided by three y times z squared times the cube root of y to the third the cube root of y to the third cancels and so that becomes y to the first and y to the first times y to the first is y squared so our final answer is two x cube root x squared y over three y squared z squared and that's it so now you know how to simplify radicals with variables and exponents so that's it for this video thanks for watching and have a have a wonderful day