You might have noticed that none of the division problems so far have had answers where a radical showed up in the denominator. We had a problem with the answer 
, but nothing with an answer like 
. This is because it works a little differently, and we haven't explained to you yet what to do with that sort of radical term. We will now, though. Hallelujah, the wait is over!
When it comes to writing fractions with radicals in them, you'll find that most teachers don't like to see radicals in the denominator. They would look at 
and be unhappy. No one wants your teacher to be unhappy, least of all your teacher. To keep your teacher happy, multiply your fraction by a clever form of 1:
Your teacher might be so thrilled that she'll decide to give everyone A's for the quarter. It might be a combination of your actions or the Lord of the Rings extended edition marathon she sat through last weekend, but either way you'll take it.
The expressions 
are mathematically equivalent. However, since radicals can be yucky (that's the technical term for it) and having radicals in denominators can be even more yucky, most people prefer the second expression.
Here's another way to see that 
and 
are equivalent, in case you're a naturally skeptical person and need more convincing. Since 
, we can rewrite like this:
Then we can cancel a factor of 
from both the numerator and denominator:
That should satisfy even the most adamant of skeptics.
Sample Problem
Rewrite the expression 
without a radical in the denominator.
First we whip out our clever form of 1. Multiplying by 
gets us to 
.
The process of rewriting so that we don't have a radical in the denominator is called rationalizing the denominator, because we're writing the denominator as a rational number instead of as some weird square root. That makes you, your teacher, Shmoop, and mathematicians everywhere happy. Everyone wins!
Before rationalizing the denominator, it's helpful to simplify the radical in the denominator, if possible.
Sample Problem
Simplify the radical term 
.
Way 1: First simplify the radical in the denominator, then rationalize. First we factor 28:
28 = 4 × 7
And then we can break up 
as:
This lets us simplify the original expression a lil bit:
From here, we can rationalize the denominator:
Way 2: This time, we rationalize the denominator before simplifying. To rationalize the denominator, we multiply by a clever form of 1.
Now we need to simplify. Since 28 = 4 × 7, we know 
, and so:
Thankfully, we got the same answer with both Way 1 and Way 2. There's also Way 3: sneak a peek down the page and steal our answer after we do all the hard work, but that won't help you much in a test-taking situation. Also, it's not as personally fulfilling.
The moral of the story is that we can rationalize the denominator first or simplify the radical first, as long as we do both before writing down our final answer. It may be more efficient to simplify the radical in the denominator first, but it's certainly not the end of the world if you don't. It's a good thing, too, because that would be far too much power for a single individual.
Of course, we can do the same sort of thing when there are variables in the radicands. Oh, you knew we were going there eventually.
Sample Problem
Simplify 
.
This seems fairly straightforward. We break up the square root into a quotient of two square roots:
It looks like we're done, but we need to eliminate that pesky radical in the denominator first. Simply shooing it away and threatening it with a radical-swatter doesn't appear to be working. To accomplish our goal, we rationalize the denominator.
In other words, we multiply by a clever form of 1:
