Rationalizing the Denominator. (Yes, We're Serious.)
Rationalizing the denominator always sounds like something that might be done at NASA just before the space station takes off. That's not the case. Allow us to break things down to size.
Rationalizing is simply the process of making sure a number is actually a rational number. A rational number is just a number that can be written as a fraction. For instance 2, ¾, 17.12, and 1,000,000 are all rational numbers. On the other hand, π and √2 are both irrational. No matter how hard we try, we can't write them as fractions.
Therefore, it would make sense that rationalizing the denominator is really just the process of writing the denominator as a rational number. The question remains: How?
Sample Problem
Simplify the following completely: 
.
To get this guy into its simplest form, we need to rationalize the denominator. Mathematicians hate, just hate, when there's a loose square root on the bottom of a fraction. Let's do them a solid and get rid of it.
Step one is to multiply our fraction by some form of 1. That form is determined by the square root in the denominator. Here it will be 
. After all, 
over is just equal to 1. See what we're talking about?
Since × is 3, we can simplify and finish up.
Don't fret. It's no sweat. Are we done yet?
Sample Problem
Simplify the following completely: 
.
Sometimes there's a bit of simplification that occurs after the denominator has been rationalized. That will happen here. But first…
After we multiply by our giant 1, we can multiply within our square roots.
At this point, we actually have two more simplifications. We know that the fraction 10⁄6 reduces to 5⁄3. But first, we can actually simplify 
.
Now we're finally ready to simplify to get our final answer.
Sample Problem
Simplify the following completely: 
.
While this problem may look complicated, our fractional exponent skills can go a long way here. Let's start by simplifying the numerator.
Next, we need to rationalize the denominator just as we've done before.
While it may not look extremely simplified, this is the best this thing is going to get.
Sample Problem
Simplify the following: 
.
This problem has a lot going on. First, we notice that we can actually solve a few of these radicals straight away. In face, we'll solve every one except for the 
.
This helps a great deal. Next, we can deal with the by writing it in simple radical form. For now, we'll leave everything else alone.
Finally, we can multiply fractions across the top and bottom while dividing out 5 over 5…
If you'd like, you can also write your final answer as 
. Either way, it's time to move on.