in this lesson we're going to focus on multiplying radical expressions so let's start with a simple example what is the square root of 3 times the square root of 5 if the index number is the same you can multiply this stuff on the inside so root 3 times root 5 is simply root 15 and we can't simplify root 15. now what about this example what is the square root of 6 times the square root of fifteen so six times fifteen is ninety and we can simplify root ninety ninety has a perfect square in it nine so let's break up ninety into nine and ten the square root of nine is three so the final answer is three root ten now granted you can simplify it before you multiply for example we can rewrite six as three and two and five i mean fifteen as three and five the square root of three times the square root of three is simply three and then we have root two times root five which we can combine as root 10 so you can also do it this way sometimes it's better to simplify before you multiply if the number is too large here's another example what about the square root of eight times the square root of six so we can multiply and say it's root forty-eight and a perfect square that goes into forty-eight is sixteen forty-eight divided by sixteen is three and the square root of 16 is 4. so the answer is 4 root 3. now what about the square root of 10 times the square root of 35. now we can multiply and get 350 but is that the best way to approach this problem 350 is a big number what we should do is break down 10 into 5 and two and thirty five into seven and five now because these two are the same we can combine them five times five is twenty five 2 times 7 is 14 and the square root of 25 is 5. so this is 5 root 14. try this one 150 times 30. so we definitely don't want to multiply right now we want to simplify before we multiply since this is going to be a very big number 150 is basically 15 times 10. thirty is fifteen times two so these two fifteen times fifteen that's uh two twenty five and two times ten is twenty the square root of two twenty five is fifteen now twenty has a perfect square in it you can break it up into four and five the square root of four is two and fifteen times two is thirty so the answer is thirty root five what about this one the cube root of eighteen times the cube root of six so we could simplify first or we can multiply together let's multiply first 18 times 6 it turns out that it's 108 and a perfect cube that goes into 108 is 27. 27 times 4 is 108 and the cube root of 27 is 3 so the answer is 3 root it's 3 cube root of 4. now we can also get the answer doing it this way as well eighteen is basically nine times two and six is three times two and nine is three times three because the index number is three to get a number outside of the radical we need three of those numbers notice that we have three cube root threes so that is just going to come out as three we don't have three twos so we can multiply the twos two times two is four and so that's another way we can get this answer now what if we have variables what is the square root of 18 x cubed times the square root of 72 x to the fifth feel free to multiply those radical expressions and simplify now what i like to do is separate the numbers from the variables 18 is basically 9 times 2. and seventy two is thirty six times two and then x cubed times x to the fifth is basically x to the eighth the square root of nine is three the square root of 36 is 6 and root 2 times root 2 that's the square root of 4 which is simply 2 and now the square root of x to the 8 is x to the fourth 6 times 2 is 12 12 times 3 is 36 so the final answer is 36 x to the fourth power now what about this one the cube root of 12 x to the fifth y to the third times the cube root of six x to the eight y to the seventh so twelve is basically two times two times three two times two is four times three is twelve and six is two times three now x to the fifth power times x to the eighth power that's going to be x to the 13th power and y to the third times y to the seven three plus seven is ten so notice that we have three twos two times two times two is eight the cube root of eight is simply two we can't take the cube root of nine so right now i'm just going to combine it and leave it as the cube root of nine now what about the cube root of x to the thirteen three goes into thirteen four times and three goes into ten three times now three times four is twelve so there's going to be one x remaining and three times three is nine ten minus nine is one so there's also one y value remaining now we can combine these two so the final answer is two x to the fourth power y to the third times the cube root of nine x y that's as far as we can go here's another problem for you multiply the four fruit of eighteen 8 to the 11 b to the 15 times the fourth root of 27 a to the 13 b to the nine so 18 we can break that up into nine times two which is basically three times three times two that's eighteen twenty seven is three times three times three now eight to eleven times eight to thirteen eleven plus thirteen is twenty-four and fifteen plus nine is also uh twenty-four now in order for a number to come out of the radical we need four of a kind we have five threes but only four of which will come out as a single three so we can take out a three and what we have left over on the inside is two times three which is 6. now 24 divided by 4 is 6. so on the outside it's going to be 8 to the 6 and b to the 6 and left over on the inside we have 2 times 3 which is 6. so this is the answer by the way if you ever want to see a question like this what would you do what is the square root of x squared plus six x plus nine feel free to pause the video and think about how we can simplify this expression notice that we have a perfect square trinomial in the form a squared plus two a b plus b squared which can be factored as a plus b squared two numbers that multiply to nine but add to six are three and three so this is going to be x plus three times x plus three which we can simply write it as x plus 3 squared and if we're taking the square root of x plus 3 squared these two will cancel and it's simply going to be x plus 3. you