Multiply and Simplify

so what I like about this problem is not only are we going to practice being able to multiply two binomials with radicals we're also going to get some practice I know you love of simplifying our radical expressions so that all comes into applying the operation and then simplifying so this is a great review to kind of go over a couple things that we need to make sure we know how to do first of all we need to apply the distributive property we have a binomial times a binomial you can see they're separated by addition and subtraction so therefore we're just going to multiply everything by by everything now of course you could go ahead and use foil whatever else but hopefully you just kind of see this diagram here by this is going to be applying the distor property now remember when you're multiplying radicals together as long as the index is the same which you can see these are all square roots all you're simply going to do is R multiply the radicand times the radicand then what we'll do is we'll combine like terms if there are any and then as well as simplify at the very very end soqu 3 * 7 that's going to be a square root of 21 a positive sare 3 times a negative so therefore that's going to be a negative square root so whatever we do we keep index the same but notice that's a positive and that's a negative so therefore that's why it's a negative these two are both positive and then we're going to multiply a 3 * a 2X Cub again you can only multiply the 3 * 2 so that's going to be a 6 x Cub then we go over here uh > of 2x * 7 that's going to be a positive < TK of 14x and then over here we're going to have square < T * negative so that's going to be negative squ root and 2 * 2 is going to be be a 4 and x * X Cub is going to be a x to 4th okay a couple things that we had and let me actually write down those rules over here so this is what we're basically working on in this case when we're multiplying our radicals now the next thing we need to understand is actually how to like simplify here the square root so just remember when you have the square root of something squared right that is just simply going to be your a all right so what we want to be able to identify is see if we can break up any of these values into square numbers or Square variables so for 21 there's no square numbers that evenly divide into 21 I can't break that up any further so therefore that is going to be non-simplified over here you might say well there's nothing I can do with a six but this is X cubed it's not x s however watch this see what I did here what I did is I rewrote an X cubed as an X2 * X now why would I do that well the cool thing is with this property just like how we can multiply the square of a * square of B to give us the square of a * B we can also take the product under a radical and separate it into their own square roots so the square of a * B is the same thing as square of a * squ of B so what that means here is I can rewrite theare < TK of 6 * x * x^ 2 I can rewrite that as thek of 6 * the of x * the < TK of x^2 now again remember what do we know about the of X2 right a s that's going to be an A so there we go we can now simplify that to X but we'll get to that in the next step over here Square 14 I can't take any square numbers to go ahead and multiply into there so therefore that is going to remain the same so Square OT of a 14x and then over here I can rewrite this as a 2^ 2 and then x to the 4th power you can all rewrite that as an X2 squar when you're simplifying the radicals with the square roots you always want to be able to see can I rewrite my radicant as a square as a square number or as a square variable and you can see I was able to do it with my variables I wasn't able to do it with my numbers but that's okay so let's go ahead and see here the 21 minus now here the S of X squ is just going to be X so that's going to be X < TK of 6X over here I'm going to have a plus Square < TK of a 14x and then over here let's see the square OT of 2^ s is going to be minus A2 and that's going to be an X2 now the last thing we want to do is be able to see can we simplify any of these and for us to be able to combine our radicals they have to not only have the same index but they also have the same radican you can see these are all different radicans so therefore there's nothing we can go ahead and simplify and that is how you multiply your two binomials with radicals