Rationalizing Denominators in Radical Expressions

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  • 0:04 Finish What Yo Start
  • 0:24 When is a Radical…
  • 1:01 Rationalizing a…
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Instructor: Jennifer Beddoe

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

Radical expressions containing denominators are not simplified completely unless the denominator is free of radical symbols. This lesson will teach you how to remove a radical from the denominator of a fraction through a process called rationalizing the denominator.

Finish What You Start

If someone was building a building, they wouldn't leave off the roof and call it good. When you make a burger, you wouldn't put all the toppings on the bun, then leave off the hamburger patty. Both of these things need to be finished in order to be usable.

Math problems also need to be finished. There are certain things that have to happen in order to call a math problem complete.

When Is a Radical Expression Finished?

There are two main requirements for a radical expression to be considered simplified as much as possible.

1. Any terms that can be removed from under the radical symbol should be removed. This means that if the radical is a square root and there is a term that can be simplified, such as a 4 or an x^2, it should be simplified. The square root of 4 is 2, and the square root of x^2 is x.

2. There cannot be a radical in the denominator of a fraction. If there is a radical in the denominator, a process called rationalizing the denominator needs to be performed.

How to Rationalize a Denominator with One Term

When the denominator is a monomial, which is an expression with a single term, you can rationalize the denominator by multiplying both the numerator and denominator by a term that will cause the denominator to be an expression so that when it is simplified it no longer contains a radical.

For example, simplify:


The next step is to determine what expression to multiply to the radical in the denominator. It needs to be something that will cause the denominator to lose its radical. Most often, the term will be equal to the term in the denominator. In the case of this example, you will multiply the numerator and denominator by the square root of 7.


When you do that, this is what happens:


Then, when you simplify and take the square root of 49, you get:


And that will be the final answer, with no radical in the denominator.

Here is another example:


This example is a bit different from the earlier one in that you don't have to multiply it by the square root of 8 in order to rationalize the denominator since 8 * 2 is 16, and 16 is a perfect square. A perfect square is a number whose square root is a whole number. We can use 2 to create a perfect square instead of 8.


When multiplied, the expression becomes:


Which can be simplified to:


as the final answer.

How to Rationalize a Denominator with More Than One Term

When there is more than one term under a radical sign in the denominator, it becomes a bit trickier. To simplify, you will need to multiply the numerator and denominator by the denominator's conjugate. The conjugate is the same expression but with the opposite sign in the middle.

The conjugate of:




When you multiply an expression containing a radical by its conjugate, this is what happens:


Remember when multiplying two binomials (expressions with two terms), you must use the FOIL method. FOIL stands for:

F = first term in each parenthesis

O = outside term in each parenthesis

I = inside term in each parenthesis

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