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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

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.

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.

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.

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.

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:

is:

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

L = last term in each parenthesis

So, for this multiplication problem, we get:

If you perform the multiplication correctly, the radicals will be eliminated. In this case:

To rationalize the denominator, you will multiply both the numerator and denominator of the fraction by the conjugate of the denominator.

Here is an example. Simplify:

First, we multiply both the numerator and denominator by the conjugate of the denominator and get:

Which gives us:

Then, we simplify by combining like terms to get:

However, we aren't done yet. This fraction can be reduced even further because each term is divisible by 2. When we cancel out a 2 from each term, the final answer becomes:

Let's try another example. Simplify:

Step one is to multiply by the conjugate of the denominator.

Which gives us:

When we simplify, we get:

Just like with building or cooking, math problems are never fully finished until certain tasks are accomplished. With fractions containing radicals in the denominator, the finishing step of rationalizing the denominator will remove that radical.

When rationalizing the denominator of a fraction, the first step is to multiply both the numerator and denominator of the fraction by a term that will cause the radical to be canceled in the denominator. In expressions where there is a single term in the denominator, this is any term that makes the denominator a perfect square. In expressions with two terms, this is the conjugate of the denominator, which is the same expression as the denominator but with the opposite sign in the middle. Once you multiply both the numerator and denominator by this term and simplify, the fraction should no longer contain a radical in the denominator.

Study this lesson as you strengthen your capacity to:

- Understand why you have to rationalize the denominator
- State the two requirements for a radical expression to be considered entirely simplified
- Walk through the steps for rationalizing denominators with one term and more than one term

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Division and Reciprocals of Radical Expressions 5:53
- Radicands and Radical Expressions 4:29
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Simplify Square Roots of Quotients 4:49
- Rationalizing Denominators in Radical Expressions 7:01
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Go to High School Algebra: Radical Expressions

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