A simplified form of a radical must meet the following three conditions

1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand does not contain a fraction.
3. No radicals are in the denominator of the fraction.

Division of radicals is really addressing the third item. No radicals can be left in the denominator. We have two types of problems. The first type has one radical term in the denominator and the second type has two terms in the denominator where one or both terms has a radical.  Both types are covered in this video. We call this process rationalizing the denominator because a radical is an irrational number, and we can’t leave it in the denominator so we rationalize the denominator by multiplying the fraction by a clever 1 (anything over itself is 1), when we do that it “pushes” the radical up to the top. It is an equivalent fraction because we have only multiplied by the number 1. Watch the video here, it covers both type of problems:

Here is the Khan Academy Lesson:

Rationalizing the Denominator

Here is a Rationalizing Higher Root Denominators example