Divide the radicands so we can cancel some stuff out:

Wow, that's like Extreme Makeover: Radical Edition. Now it makes sense to break up the radical again:

.

Finally, we need to rationalize the denominator. Multiply by to find

.

Example 2

Divide: .

We break up the fraction into two fractions:

, we can simplify the first term:

Rationalize the denominator on the second term, and we have our final answer:

.

Example 3

Divide: .

Folks, we have ourselves a quotient here! The denominator has two terms and contains a radical, which is exactly what we were looking for. To rationalize this denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is .

Here we go:

Since the denominator is only a number now and a shell of its former self, we can break up the fraction:

.

That broke up quicker and easier than your AT&T cell phone connection.

Example 4

Divide: .

The conjugate of the denominator is , so we multiply this by the numerator and denominator:

There are no radicals in the denominator at this point, so we're done.