Rationalizing Denominators

 

Learning Objective(s)

·         Rationalize a denominator with a monomial containing a square root.

·         Rationalize a denominator that contains two terms.

 

 

Although radicals follow the same rules that integers do, it is often difficult to figure out the value of an expression containing radicals. For example, you probably have a good sense of how much  and  are, but what about the quantities  and ? These are much harder to visualize.

 

That said, sometimes you have to work with expressions that contain many radicals. Often the value of these expressions is not immediately clear. In cases where you have a fraction with a radical in the denominator, you can use a technique called rationalizing a denominator to eliminate the radical. The point of rationalizing a denominator is to make it easier to understand what the quantity really is by removing radicals from the denominators.

 

 

The idea of rationalizing a denominator makes a bit more sense if you consider the definition of “rationalize.” Recall that the numbers 5, , and  are all known as rational numbers—they can each be expressed as a ratio of two integers (, and  respectively). Some radicals are irrational numbers because they cannot be represented as a ratio of two integers. As a result, the point of rationalizing a denominator is to change the expression so that the denominator becomes a rational number.

 

Here are some examples of irrational and rational denominators.

 

 

Now let’s examine how to get from irrational to rational denominators.

 

 

Let’s start with the fraction . Its denominator is , an irrational number. This makes it difficult to figure out what the value of  is.

 

You can rename this fraction without changing its value, if you multiply it by 1. In this case, set 1 equal to . Watch what happens.

 

 

The denominator of the new fraction is no longer a radical (notice, however, that the numerator is).

 

So why choose to multiply  by ? You knew that the square root of a number times itself will be a whole number. In algebraic terms, this idea is represented by . Look back to the denominators in the multiplication of . Do you see where ?

 

Here are some more examples. Notice how the value of the fraction is not changed at all—it is simply being multiplied by another name for 1.

 

 

 

 

You can use the same method to rationalize denominators to simplify fractions with radicals that contain a variable. As long as you multiply the original expression by another name for 1, you can eliminate a radical in the denominator without changing the value of the expression itself.

 

 

 

 

 

 

Rationalize the denominator and simplify.   A) B) C) D)   Show/Hide Answer A) Incorrect. After you rationalize a denominator, the denominator should not be in radical form. Try multiplying the fraction by . The correct answer is .   B) Incorrect. The process of rationalizing does not remove the denominator—it just changes the fraction so that the denominator is no longer a radical. Try multiplying the numerator and denominator by . The correct answer is .   C) Incorrect. You may have rationalized the fraction  instead of . Try multiplying the numerator and denominator by . The correct answer is .   D) Correct. To rationalize the denominator, multiply  by . This will create an integer in the denominator, so the fraction will have been rationalized.

 

 

 

Denominators do not always contain just one term, as shown in the previous examples. Sometimes, you will see expressions like  where the denominator is composed of two terms,  and +3.

 

Unfortunately, you cannot rationalize these denominators the same way you rationalize single-term denominators. If you multiply  by , you get . The original  is gone, but now the quantity  has appeared...this is no better!

 

In order to rationalize this denominator, you want to square the radical term and somehow prevent the integer term from being multiplied by a radical. Is this possible?

 

It is possible—and you have already seen how to do it!

 

Recall that when binomials of the form  are multiplied, the product is . So, for example, ; notice that the terms −3x and +3x combine to 0. Now for the connection to rationalizing denominators: what if you replaced x with ?

 

Look at the side by side examples below. Just as  combines to 0 on the left,  combines to 0 on the right.

 

 

 

There you have it! Multiplying  by  removed one radical without adding another.

 

In this example,  is known as a conjugate, and  and  are known as a conjugate pair. To find the conjugate of a binomial that includes radicals, change the sign of the second term to its opposite as shown in the table below.

 

 

 

 

 

 

 

One word of caution: this method will work for binomials that include a square root, but not for binomials with roots greater than 2. This is because squaring a root that has an index greater than 2 does not remove the root, as shown below.

 

 

 

 cannot be simplified any further, since its prime factors are . There are no cubed numbers to pull out! Multiplying  by its conjugate does not result in a radical-free expression.

 

 

Identify the conjugate of the denominator.   A) B) C) D)   Show/Hide Answer A) Correct. The conjugate will be the binomial that, when multiplied by the denominator, eliminates the radical. The conjugate is .   B) Incorrect. The denominator is not the conjugate. Look for the binomial that, when multiplied by the denominator, eliminates the radical. The correct answer is .   C) Incorrect. Multiplying the denominator by  will not remove the radical. Look for the binomial that follows the pattern . The correct answer is .   D) Incorrect. Multiplying the denominator by  will not remove the radical. Look for the binomial that follows the pattern . The correct answer is .

 

 

 

When you encounter a fraction that contains a radical in the denominator, you can eliminate the radical by using a process called rationalizing the denominator. To rationalize a denominator, you need to find a quantity that, when multiplied by the denominator, will create a rational number (no radical terms) in the denominator. When the denominator contains a single term, as in , multiplying the fraction by  will remove the radical from the denominator. When the denominator contains two terms, as in, identify the conjugate of the denominator, here, and multiply both numerator and denominator by the conjugate.