Dividing Radical Expressions

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  • 0:03 The Quotient Rule
  • 1:07 Dividng Radical Expressions
  • 2:06 Rationalizing the Denominator
  • 3:59 Putting It All Together
  • 7:11 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

When dividing radical expressions, we use the quotient rule. This lesson will describe the quotient rule and how to use it to solve these radical expressions.

The Quotient Rule

A quotient is the answer to a division problem. When dividing radical expressions, we use the quotient rule to help solve them. The quotient rule states that a radical involving a quotient is equal to the quotients of two radicals. Written out in math terms, this means:


So when you divide one radical expression by another, you can simplify it by writing both expressions under the same radical, then simplifying. The quotient rule works only if:

1. Each radical has the same index. The index is the superscript number to the left of the radical symbol, which indicates the degree of the radical.


In this example, the index is the 3 and it is indicating the cube root of 27. If there is no index, it is understood to be 2, or a square root.

2. The denominator of the fraction is not zero.

Dividing Radical Expressions

Here are the steps to dividing radical expressions.

  1. Ensure that the index of each radical is the same and that the denominator is not zero.
  2. Convert the expression to one radical.
  3. Simplify where possible.
  4. Rationalize the denominator, if necessary.

For example, solve:


Once we have determined that both the numerator and denominator have the same index (2) and that the denominator is not zero, we can use the quotient rule to convert the expression to this:


Solving under the radical - 100/4 = 25 - gives us √25, which is equal to 5. Since there is no denominator to rationalize, the problem is finished.

Rationalizing the Denominator

Mathematics has its own language, and like any other foreign language, there are certain rules that must be followed to speak the language correctly. Speaking math correctly ensures that mathematicians all around the world are able to understand each other.

One rule for radicals is that for a radical expression to be in its simplest form, it cannot have a radical in the denominator. So the last step in simplifying any radical expression with a fraction is to make sure that there is not a radical in the denominator. This is called rationalizing the denominator.

To rationalize a denominator, you multiply the expression by an appropriate fraction that is equivalent to one. For example, simplify:


Because there is a radical in the denominator of this expression, we need to rationalize the denominator to simplify the expression so it is written mathematically correct. The first step is to create a fraction that is equivalent to one that will help remove the radical from the denominator. Most of the time, the fraction needed will be the same as the radical in the denominator. To make it a fraction that is equivalent to one, you must have the same numerator as denominator. For this example, the first step of the problem will look like this:


The simplification of this problem looks like this:



Because the square root of x^2 is equal to x, the radical is removed from the denominator and the simplification of this expression is


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