Everything you've learned about rationalizing the denominator goes out the window if the denominator of your fraction has a binomial in it. For example, if the denominator is 3 + , you might be tempted, based on what you know, to multiply the numerator and denominator by ; after all, that's the radical in the denominator, right?

But look what happens if you multiply 3 + by :

(3 + ) =
3 + ² = 
3 + 4

That's still not rationalized. You could keep multiplying by over and over again, but it would never become rationalized.

So what do we do?

Well, we need to remember one of our nice rules for factoring/multiplying binomials. It's the difference of squares rule, and it looks like this:

(a + b)(a - b) = a² - b²

Why does that help us? Well, if we multiply (3 + ) by (3 - ), then we'll end up with 3² - ², which is 9 - 2, or 7. And presto! Our radical is gone!

So let's try an example of this: Simplify .

To rationalize the denominator, we need to multiply the numerator and denominator by (5 + ).

=  · = = =

Of course, the problem will be slightly more challenging if there's also a binomial in the numerator: Simplify .

Since the denominator is (2 - ), we can simplify by multiplying the numerator and denominator by (2 + ).

=  · = = = 3 + 2

1.

Simplify

2.

Simplify

3.

Simplify

4.

Simplify

5.

Simplify

6.

Simplify

7.

Simplify

8.

Simplify +