We already know how to divide one radical term by another, and how to simplify our answer by rationalizing denominators. However, because you looked sleepy the last time we went over it, let's review.
Sample Problem
Divide: 
.
Since 5 divides nicely into 25, we can divide the radicands:
There are two other kinds of division with radical expressions that you're likely to be asked about by a teacher, or on a test, or even by a curious stranger on the street: dividing a multinomial by a single term, and dividing a multinomial or binomial by a binomial. Adjust your goggles and take a deep breath, because we're diving in.
Dividing a multinomial by a single term is similar to dividing a polynomial by a single term. We break up the quotient into several simpler quotients, simplify those, and add them together. This isn't our own idea; we're just trying to maintain the status quotient. Ba dum tsch!
Sample Problem
Divide: 
.
Break up the quotient into several nicer quotients. If you're using a hammer, watch your thumbs.
We can simplify each of these quotients:
Adding the simplified forms gives us:
Now we're almost done. Hey, the first and last terms have the same radicand, so we can group 'em together:
Now we're done. We promise. Pinky swear. No, seriously, put your pinky up to the monitor. Then, wait until we come out of the computer to meet you.
Dividing by a binomial is a little weirder. Not like Lady Gaga weird, though; maybe only Tom Cruise weird. We'll show how this works with an example.
Sample Problem
Divide: 
.
Multiply by a clever form of 1. We just learned this, and this is crazy, but trust us. It will work, and we'll explain why we chose this crazy-looking clever form of 1 after the example. Hang in there! And call us maybe? On second though, don't. You have algebra to learn.
Now we multiply things out. Hold on, things will get scary for a second.
Notice what happened in the denominator: we multiplied two binomials of the form 
.
After multiplying, the radical went away because we had a difference of two squares.
Radical expressions of the form are called conjugates. Go ahead and file that away for future reference. Preferably under "C," so you'll be able to find it when you need it.
Sample Problems
- The conjugate of
is 
.
- The conjugate of
is 
.
- The conjugate of
is 
.
When we have a quotient of expressions where the denominator has two terms and contains at least one radical, we can rationalize the denominator, or eliminate the radical(s) in the denominator. (We think "rationalizing" something sounds better than "eliminating" it. What can we say, we're softies like that.) We rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
Or, in other words: put that sign down, flip it, and reverse it.