Wow, 18 replies and not a single attempt at an answer... Come on, Internet, we can do better than this.
Anyway, there are a lot of irrationals in that denominator, but there should be a way to make them all go away. First we'll try the conjugate rule.
The conjugate rule works like this: If you have something like 1/(√6+√5), you multiply the top and bottom by (√6 - √5), the conjugate. This works with the FOIL method, or First-Outside-Inside-Last.
Start with (√6+√5)(√6-√5). Take the First term from each set of brackets and multiply them. . . . . √6 x √6 = 6
. . . . . . . . . . . . . . . . . . . Then the Outside term from each set of brackets. . . . . . . . . . . . . . . . √6 x -√5 = -√30 (The negative is important!)
. . . . . . . . . . . . . . . . . . . Then the Inside. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √5 x √6 = √30
. . . . . . . . . . . . . . . . . . . Then the Last. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √5 x -√5 = -5
When you add them all together, you get a nice, simple, 1. Just 1. So now the fraction is (√6 - √5)/1.
The complicated part about your problem is that you have three square roots instead of two. We can still use the conjugate rule, it's just going to be a little trickier.
First, we decide where we split our denominator. I'm going to treat it as (√2 + √3) + √5. This is the same thing we had before, just organized differently. So our conjugate is going to be (√2 + √3) - √5.
Now we use the FOIL rule, which gets more complicated again... but it still works. You just have to use it twice, at the same time. I'll show you what I mean:
Multiplying ((√2 + √3) + √5) x ((√2 + √3) - √5)
First - (√2 + √3) x (√2 + √3) This is where we use FOIL again. I'm not going to show you each step, but it works out to 2√6 + 5.
Outside - (√2 + √3) x -√5 = -√10 - √15
Inside - (√2 + √3) x √5 = √10 + √15
Last - √5 x -√5 = -5
So, adding all that together, the fraction becomes (√2 + √3 - √5) / (2√6). Usually, this will leave you with another number still on the bottom, but then you can use the conjugate rule (I know, again?) to get rid of it. In this case, though, simply multiply the top and the bottom by √6.
I'll skip a few steps and tell you that the answer is (√12 + √18 - √30) / 12.
Now, always remember how to use FOIL. It's not just for square roots: the basic form applies to everything you could possibly multiply together. Happy homeworking!
