what's up guys in this video i want to be able to help you add and subtract radicals but not just be able to do the problems actually have an understanding of when why and how we can go ahead and do about that so by the end of this video i'm going to be able to show you how to combine this equation as well as an expression like this before we actually get into adding and subtracting radicals i think it's really important for students to be able to understand some of the basics right so let's go and start with the basics real quick and i want to explain these basics so you have a better understanding of why we add radicals the way that we do so let's just do something that i think we can all agree to so if i had 5 plus 5 i think this is relatively easy for most students they recognize hey that's going to equal 10 right but there's another way we could write 5 plus 5 and this is very important so we're doing is we're grouping this so what's really important is i can also write this as a 2 times 5. i can group the 5 because not only are the 5's both numbers right but they're also the exact same number so rather than just combining them 5 plus 5 i can also group them and when we're thinking about adding and combining like terms that's the idea we want to look for is how can we group them together now i know that 5 and 6 are both numbers right and so we can combine them 11. however i want you to understand we cannot group them like we did over here right this does not equal a 2 times you know 5 or it does not equal a 2 times 6. so here we come to a problem even though they're both numbers we cannot group them this is the really important thing we need to understand here because for instance like apples and oranges are both fruit but we can't combine them right an orange plus an orange is two oranges an apple plus an apple is two apples but when you have an orange and an apple you can't say that's two apples or two oranges right you could say that's two fruit so semantically we can work around that but when we're dealing with math and variables it's very important to recognize how we can go ahead and combine things so x plus x is yes that is 2x but it's very important that we can recognize this to be a 1 plus 1 right of the x's so you can see here how i am grouping them however if i had an x plus y i can't combine these any other way right so therefore these are not like terms and then the same thing if i had an x squared plus an x squared so if we have exponents well i can rewrite that as again a 1 plus 1 of the x squared again i am combining these because they're both x squared but if i had an x squared plus a y squared well i cannot combine these any further if i have a let's say a square root of x plus a square root of x now again these are both square roots of x right they're exactly the same so therefore i can combine them as a 1 plus 1 times the square root of x however if i have the square root of x plus the square root of y i cannot combine those or if i have the square root of x plus the cube root of x i cannot combine these so it's very important when we're looking at all these different examples the only thing that we could truly combine is when they are exactly the same so not just that they're both numbers or that they're both variables right or they're both exponents or they're both radicals but when they are exactly the same and what we're going to focus on in this lesson is when these are exactly the same right not only do you have to have the radicand what is under the radical being the same but you also have the index this is the cube root this is the square root so if not everything's not the same we cannot simply combine them so this is the idea and the vision that i want you to come through when we start working through some problems let's go and take a look at this first example and kind of see how we can apply this understanding all right so in this example you can see we have square root of 18 plus the square root of 32. now the cool thing is the index is the same right these are both square roots even though a 2 is not written in there we both have a 2 there now the cool thing is the radicands are not the same so what i'm going to want to do is look into simplifying and if you don't know how to simplify radicals then go and check out my previous video on simplifying radicals so now assuming that you have a basic background of simplifying radicals what i'm going to want to do is rewrite these as a product using square numbers and the largest square number that you can think of so in this case when i see 18 i recognize this i can rewrite this as a 9 times 2. and for the square root of 32 i recognize this to be a 16 times 2. and again this is very important because 9 and 16 are square numbers so again like you don't have to write it this way but again remember the rules of radicals we can break this up right so we can rewrite that as a product and that's important because here i know i can simplify the square root of 9. that's going to be 3 square root of 2 plus the square root of 16 is going to be a 4 square root of 2. and now what i want you to recognize here is my radicands are exactly the same my index is exactly the same so now i can combine them and what i'm saying by that is i can just write a 3 plus 4 of a square root of 2 right and 3 plus 4 is going to be a 7 square root of 2. so we're using that same ideas once you have the exact same elements you just combine how many of them you have right in my previous examples we just had one of them so it was 1 plus 1 right but in this case we have three square root of twos plus four square root of twos but since they're exactly the same we can now go ahead and add them okay so in this next example it's gonna be a little bit different now we have some numbers in front of our radicals but again that doesn't really matter when we're looking into combining our radicals we just want to make sure they're exactly the same having the same index which is cool we have a 3 here as well as having the same radicand which is not cool here because we don't have the same radicand so now what i want to be able to do is again now we got to simplify our cube roots and again if you don't know how to simplify cube roots go ahead and check out my previous video i have for you there now in this example there's not really much we actually have to do there's nothing i can simplify here now in this example we want to find the largest cube number that evenly divides into 40. and remember our smallest two cube numbers we have is 8 and 27 and i recognize 8 evenly divides into 40. so again using my rules of radicals or just using a product i'm going to break this up as an 8 times 5. and now again you know you can simplify that if you like to just do a little bit extra work here i can rewrite that as a cube root of 8 times the cube root of 5. and again the reason why i am simplifying this is again this negative 4 is being multiplied by the cube root of 8. so i'm going to rewrite this like kind of step by step so you can see this but this is negative 4 and then this is going to be times a 2 times a cube root of 5. now you can see here i have a 7 cube root of five now this becomes a minus eight cube root of five we have the exact same radicals both the cube root and both of five so now i can just go ahead and say seven minus eight of the cube root of five and seven minus eight is going to be a negative 1. so that'd be a negative 1 times the cube root of 5 which we can just write as a negative cube root of 5. so there it goes ladies and gentlemen i hope this was helpful for you if you want more examples of combining radicals go and check out the examples i have for you down below or check out my next video i have for you here cheers