Dividing Radicals and Rationalizing the Denominator - Problem 3

I need to simplify this fraction and the way I’m going to do it is by rationalizing the denominator because I have the sum of 2 radicals and a denominator.

Rationalizing the denominator means I’m going to multiply the top and bottom by the conjugate of this guy. That’s a fancy word for just changing the sign there.

Root 12 minus root 8 gets multiplied on top and bottom. On top let’s go ahead and distribute I’ll have -3 root 12 plus 3 root 8, careful with the minus signs that’s what tripped me up minus, minus is positive. On the bottom let’s FOIL, root 12 times root 12 is regular 12. My outers I would have negative root 8, what’s 8 times 12? 96 right 96, sorry. I would have negative root 96 plus root 96, those guys cancel out, and then when I look at my lasts I’m going to have minus 8 there. That’s great my denominator no longer has any radical expressions that means I’m doing this process correctly.

The whole point of multiplying by the conjugate was to get rid of the radicals down here. So now the bottom of my fraction is just 4, on top though I’m going to have to do some simplifying. The square root of 12 I’m going to simplify that as the square root of 4 times square root of 3 which is 2 root 3. So I have 2 root 3 times negative 3 which is going to be -6 root 3 again because I did -3 times 2 to get -6 then I had root 3 left that’s my first term.

Here I can also reduce square root of 8. Square root of 8 is the same thing as square root of 4 times square root of 2 or 2 root 2. So now I have 3 times 2 root 2 which is going to be 6 root 2. This can be simplified further check it out. I have a common factor. I have 6 or -6 it’s up to you which one you choose to factor out. I’m going to factor out +6, you might also choose to factor out -6, you should get the same answer. I factor out +6 then I’ll have negative root 3 plus root 2 on top, bottom I have the 4.

Finally my last step is going to be to reduce 6 over 4 as 3 over 2. This will be my final answer. 3 times the quantity negative root 3 plus root 2 on top of 2. That was tricky I had to use lots of different skills and that’s why Math teachers love to assign problems like these. You have to know how to find the conjugate, you have to be able to multiply correctly so that your denominator no longer has rational terms, you have to be able to reduce both of these guys, be careful with the signs, and also remember to multiply by these 3s, then you have to factor out a common factor and finally recognize that the fraction can be reduced.

This is kind of a nasty problem but I bet your cruel Math teacher is going to make you do something like this. You can do it just keep track of all your positives and negatives. Be really careful to make sure it's all the way simplified and you guys will get these ones right.