less than ten three operations with radical expressions in this video you will learn to add in subtract radical expression and multiply radical expressions so first thing we're going to do is we're going to look at this and they have the same radicand so remember the radicand is part inside the square root so in here all of these have a 5 in the radicand which means there's like terms so I want to add and see how many square root of fives I have two of them here and negative five of them they're 6 and 2 makes 8 and then 5 less makes a total of 3 square root of 5 it's almost like you're thinking about square root as it was like a variable X so like if you had 6x plus 2x minus 5x you would say that that is 3x it's the same thing except instead of X we have a square root and that's what we do when we have the same thing the same term so looking at this square root 2 and here we have a square root 2 so those are our two like terms so 7 square root 2 minus 6 square root 2 7 minus 6 is 1 so I'm just left with square root 2 and a square root of 11 so 8 and negative 4 makes a total of positive 4 square root 11 and now I have to simplify it I can go because 2 and 11 both can't simplify any further so and they're not eye terms so they can't be combined what if I have all different square roots so right now looking at this it looks like they're all different terms and they can't be combined but what I want to do first is go through in December so I'm asking myself are there any perfect squares that go into these so into 27 nine times three is 27 so I can say does this think they six times the square root of nine times three which is going to become six square root of 9 is 3 so 6 times 3 is 18 and I still have that spirit 3 on the inside now I'm looking at the next one 8 plus the square root of 12 well the square root of 12 inside 12 we can say that the same thing is 4 times 3 and now I can think is its square root of 4 so square root of 4 is 2 times that 8 this is going to become 16 square root 3 and then a 75 five goes into that so I can say this is the thing or 25 I mean this 2 times the square root of 25 times 3 so I can say this is thinking it's square root 25 which is 5 times 2 is 10 square root 3 and now if I'm looking at each of my radicand they're all great which means that these are all like terms so plus a plus 10 16 and 18 is 24 plus 10 makes a total of 34 square root 3 and now that's the final answer okay if I want to multiply I multiply the numbers first so 2 and 4 makes 8 and then I multiply the square root so 3 and 6 is 18 now in 18 a perfect square of 9 can be written so this is 9 times 2 square root of 9 is 3 so I have 8 times 3 times square root of 2 8 times 3 is 24 square root 2 and that's all for this one I want to use the distributive property so 4 square root 2 times 3 square root 2 multiplied 4 times 3 and I get 12 square root 2 times square root 2 makes the square root of 4 so 4 times that is eight and spirit two times square root 6 makes well and then I wanted to simplify so the square root of 4 12 is 24 and then over here I can say this is same thing as 8 times 4 times 3 + square root of 4 is 2 times 8 so I get 24 plus 18 times the spirit 3 and now that's in simplified form okay this next one find the area of a rectangle in simplest form with a width of that and a length of that thing right so the area of a rectangle is length times width so I want to multiply my length times my width of my leg it's 5 square root 3 plus 7 square root PI and my width is 4 square root 6 minus 2 square root 10 so when I have a binomial times a binomial I'm using the foil method so my first 3 time is 4 squared 6 so 5 times 4 is 20 and square root 3 Times Square six makes square root of 18 if I do my outside terms 5 square root 3 and negative 2 square root 10 negative 2 and 5 makes negative 10 squared 3 times per 10 is spurred 30 inside terms 4 times 7 is 28 and 5 times 6 is 30 and then my last term's 7 times negative 2 is negative 14 and 5 times 10 is the square root of 50 so 20 square root 18 amazing thing is 20 times the spirit of 9 times 2 minus 30 in [Music] we can combine these ten and twenty eight positive twenty and a negative ten will make positive 18 square root 30 and then 50 we can say that the same thing is 25 times 2 so then simplifying square root of 9 is 3 times 20 is 60 square root 2 plus 18 square root of 30 and then minus square root of 25 is 5 times 14 is 70 so this is going to be minus 70 square root and that's a simplified it both sorry not like 60 and negative 70 heroes for twos so I combined them and I can say that this is 18 square root 30 minus 10 square root 2 okay so that is example 4 so just a summary um some things you can do with radicals if you have addition so you have a square root B plus B Square B feed square of B sorry you can pull out the scurvy and say the same thing as a plus C it's very P so that's what we were doing before like if I had 3 square root 2 plus 2 square root 2 we were special essentially saying ok we're doing 3 plus 2 times the square root of 2 we are combining like terms the saying that this was 5:30 so this is just a few extra steps than what we were doing ok we can do the same thing with subtraction and then with multiplication we multiply the coefficients together so that's the numbers in front and then we will heterotic hands together and put them inside them so the root and then we simplify anything that we can and that is 10 3