Learning Objectives
Introduction
In this tutorial we will talk about rationalizing the denominator and numerator of rational expressions. Recall from Tutorial 3: Sets of Numbers that a rational number is a number that can be written as one integer over another. Recall from Tutorial 3: Sets of Numbers that an irrational number is not one that is hard to reason with but is a number that cannot be written as one integer over another. It is a non-repeating, non-terminating decimal. One example of an irrational number is when you have a root of an expression that is not a perfect root, for example, the square root of 7 or the cube root of 2. So when we rationalize either the denominator or numerator we want to rid it of radicals.
Tutorial
Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth.
Keep in mind that as long as you multiply the numerator and denominator
by the exact same thing, the fractions will be equivalent.
Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.
So in this case we can accomplish this by multiplying top and bottom
by the square root of 6:
*Den. now has a perfect square under sq. root
AND
Step 3: Simplify the fraction if needed.
*Sq. root of 36 is 6
*Divide out the common factor of 2
So in this case, we can accomplish this by multiplying top and bottom
by the cube root of :
*Mult. num. and den. by cube root of
*Den. now has a perfect cube under cube root
AND
Step 3: Simplify the fraction if needed.
*Cube root of 27 a cube is 3a
Also, we cannot take the cube root of anything under the radical.
So, the answer we have is as simplified as we can get it.
Rationalizing the Numerator
(with one term)
Note that these are the same basic steps for
rationalizing a denominator, we are just applying to the numerator now.
Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth.
Keep in mind that as long as you multiply the numerator and denominator
by the exact same thing, the fractions will be equivalent.
So in this case, we can accomplish this by multiplying top and bottom
by the square root of 5:
*Mult. num. and den. by sq. root of 5
*Num. now has a perfect square under sq. root
AND
Step 3: Simplify the fraction if needed.
*Sq. root of 25 is 5
Also, we cannot take the square root of anything under the radical.
So, the answer we have is as simplified as we can get it.
So in this case, we can accomplish this by multiplying top and bottom
by the cube root of :
*Num. now has a perfect cube under cube root
AND
Step 3: Simplify the fraction if needed.
*Cube root of 8 x cube is 2x
Also, we cannot take the cube root of anything under the radical.
So, the answer we have is as simplified as we can get it.
Rationalizing the Denominator
(with two terms)
Because we now have two terms, we are going to have to approach it differently
than when we had one term, but the goal is still the same.
a + b and a - b are conjugates
of each other.
So what would the conjugate of our denominator be?
It looks like the conjugate is .
AND
Step 4: Simplify the fraction if needed.
So what would the conjugate of our denominator be?
It looks like the conjugate is .
AND
Step 4: Simplify the fraction if needed.
*12 is (4)(3) and sq. root of 4 is 2
*18 is (9)(2) and sq. root of 9 is 3
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problem 1a: Rationalize the Denominator.
Practice Problem 2a: Rationalize the Numerator.
Practice Problem 3a: Rationalize the Denominator.
Need Extra Help on these Topics?
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 21, 2011 by Kim Seward.
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