in this lesson we're going to focus on simplifying radicals so let's say if we have the square root of 36 the square root of negative 49 negative square root 64. and negative square root negative 25. how would you simplify these four radicals the square root of 36 is simply 6 it's a real number the square root of a negative number is not a real number it's an imaginary number so this is equal to seven i where i is the square root of negative one if you have it learn about imaginary numbers uh you cannot worry about it so this doesn't give you a real number it gives you an imaginary number but the way you show your work though here's what you can do first separate the 49 and the negative one the square root of 49 is 7 and the square root of negative 1 is i so you get 7i the square root of 64 is 8 but there's a negative in front so it's negative 8. now for this one we have negative square root 25 times the square root of negative 1. the square root of 25 is positive five and the square root of negative one is i so the final answer is negative five i now you need to be familiar with the perfect squares one squared is one two squared is four three squared is nine four squared is sixteen five squared is twenty five and then thirty six forty nine sixty four eighty one and a hundred i'm only going to go up to 10 this video but you may want to know it up till 20. 20 squared is 400 now let's say if you want to simplify the square root of 75 would you do now this is not a perfect square but what i recommend doing is to break down 75 into two numbers one of which is a perfect square so looking at the list of numbers that you see here which of these numbers go into 75 75 is a multiple of 25. if you divide 75 by 25 you get 3 so 25 times 3 is 75 now the square root of 25 is 5. and so that's how you could simplify a radical like this the square root of 75 is equal to 5 square root 3. now let's look at some more examples because you want to master this technique so i'm going to give you a lot of examples to practice on go ahead and simplify the square root of 18 and the square root of 48 the best way to learn is by practice so you may want to pause the video and try these problems yourself to make sure you understand this concept so what perfect square goes into eighteen nine goes into eighteen two times so i'm gonna write nine times two the square root of nine is three so the square root of eighteen simplifies to three square root two now let's do the same for forty-eight what perfect squares go into forty-eight 4 can go into it and 16 can go to it when you have two perfect squares that can go into 48 you want to choose the larger of the two values 48 divided by 16 is 3. so you want to break down 48 into 16 and 3. the square root of 16 is 4 and so it simplifies to 4 square root 3. try these two problems simplify 8 square root 80 and also 5 square root 98 so the square root of 80 how can we simplify it if you're not sure what to do take 80 and divide it by each perfect square 80 divided by 4 is 20. 9 does not go into 80. 80 divided by 16 is 5. and 25 doesn't go into 80. so the highest perfect square that goes into it is 16. so i'm going to write the square root of 80 as the square root of 16 times the square root of 5. now the square root of 16 is 4. and now at this point all we need to do is multiply 8 by 4 which is 32 so the answer is 32 square root 5. now for the next one what perfect square goes into 98 49 is half of 98 so we can write it as 49 times 2. the square root of 49 is 7 and 5 times 7 is 35 so we have 35 square root 2 and this is the answer now if you're asked to simplify this fraction 5 divided by the square root of 2 what can you do all you can do in this problem is rationalize the denominator and to do that multiply the top and the bottom by the square root of two so on top you can have five square root two on the bottom the square root of two times the square root of two is the square root of four which is two so your final answer is five square root two divided by 2. so for the sake of practice try these two examples so for the first one let's multiply the top and the bottom by the square root of 3. the square root of 3 times the square root of 3 is the square root of 9 which the square root of 9 is 3. so this is the answer and so for the last one we're going to do the same thing now the square root of 5 times the square root of 5. we know in the end it's going to be just 5. the square root symbols will cancel and so you get that as the answer now you need to know what to do when adding and subtracting radicals so for example how can we simplify this expression 4 square root 8 plus 3 square root 15 minus 6 square root 32 now think about how we simplify the square root symbols or the radicals individually think of a perfect square that goes into eight four is the highest perfect square that goes into eight so i'm going to write eight as four times two twenty-five is a perfect square that goes into fifty so i'm going to replace 50 with 25 and 2 and 16 goes into 32 so i'm going to write it as 16 times 2 now the square root of 4 is and the square root of 25 is 5. the square root of 16 is 4 and now we need to multiply 4 times 2 is 8. 3 times 5 is 15. and six times four is twenty four so now we can add the coefficients eight plus fifteen is twenty three and twenty three minus 24 is negative one so it's negative one square root two which is the same as negative square root two so this is the final answer here's another example for you 7 square root 27 plus 3 square root 12 minus 5 square root 48 go ahead and simplify this problem so 9 is a perfect square that goes into 27 27 is 9 times 3 4 is the perfect square that goes into 12 and so 4 times 3 is 12. and 48 is 16 times 3. so it helps to know your multiplication values now the square root of 9 is 3 and the square root of 4 is 2. the square root of 16 is 4. next multiply 7 times 3 is twenty one three times two is six five times four is twenty and now let's combine the coefficients so twenty one plus six is twenty seven and 27 minus 20 is seven so the final answer is seven square root three so now you know how to add and subtract radical expressions now let's say if you have this expression 8 divided by 3 square root 2. how can we simplify the expression if you have a fraction with radicals it might be wise to multiply the top and bottom by the conjugate in this case of the denominator so the conjugates can be the same thing but instead of a minus we're going to have a plus in between the three and the square root of two so on the top we gotta distribute the eight eight times three is twenty four and eight times the square root of two is simply eight root two on the bottom we need to foil three times three is nine and then we have three times the square root of two and three times negative square root two and then negative square root two times square root two is just negative two now the two middle terms three and negative three adds up to zero so right now we're left with 24 plus 8 square root 2 divided by 9 minus 2 which is 7. so you can leave your answer like this or you could separate it into two fractions you can write it as 24 divided by 7 plus 8 over 7 square root 2 if you want to but that's how you could simplify the expression that we had let's work on another example simplify this one 3 plus the square root of 2 divided by 5 minus the square root of 2. so what should we do the best thing to do is to multiply by the conjugate of the denominator that is five plus the square root of two so let's foil on top so we have 3 times 5 which is 15 and then we have 3 times the square root of 2 plus 5 times the square root of 2 and then the square root of 2 times the square root of 2 that's just 2. on the bottom we have 5 times 5 which is 25 and then 5 times the square root of 2 and then 5 times negative square root 2 and finally negative square root 2 times square root 2 which is just negative 2. so in the denominator 5 and negative 5 adds up to 0. so they're going to disappear and then we can combine 15 plus 2 which is 17. and we could also combine 3 plus 5 since they share the same radical so 3 plus 5 is 8 so we have 8 square root 2. on the bottom we have 25 minus 2 which is 23. so the final answer we can write it as 17 divided by 23 plus 8 over 23 times the square root of 2. you