Chapters
In this article, we will discuss how to rationalize denominators with radicals.
What is Rationalization?
"The process of removing radicals or imaginary numbers from the denominator in such a way that the number in the denominator is converted to a rational number only is known as rationalization"
Removing the radicals from the denominator of the fraction allows us to add, subtract, multiply and divide the fractions in a simple way.
There are three cases when we have a radical in the denominator. We follow different rationalization procedures to remove the radicals in all these three cases. These three scenarios are:
- When we have a fraction of the type
- When we have a fraction of the type
- When we have the fraction
In the next section, we will solve examples in which we will rationalize the fraction belonging to the above three categories.
Example 1
Solve 
.
Solution
To rationalize the above fraction, we need to multiply and divide the fraction by the root in the denominator. In the above example, the root in the denominator is 
. Hence, we will multiply and divide by like this:
=
= 
= 
= 
= 
is equal to 
. Hence, we can simplify the above expression further like this:
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Example 2
Solve 
.
Solution
To rationalize the above fraction, we need to multiply and divide the fraction by the root in the denominator. In the above example, the root in the denominator is 
. Hence, we will multiply and divide by like this:
=
= 
= 
= 
= 
Example 3
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by 
. In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
= 
=
= 
Example 4
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by . In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
= 
= 
= 
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Example 5
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by . In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
= 
= 
= 
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Example 6
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by . In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
= 
= 
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Example 7
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by . In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
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= 
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Example 8
Rationalize the following fraction:
Solution
When we have the fraction of the type 
, then we rationalize the denominator by multiplying and dividing the numerator by 
.
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Example 9
Rationalize the following fraction:
Solution
When we have the fraction of the type , then we rationalize the denominator by multiplying and dividing the numerator by .
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Example 10
Rationalize the following fraction:
Solution
When we have the fraction of the type 
, then we rationalize the denominator by multiplying and dividing the numerator by 
.
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Example 11
Rationalize the following fraction:
Solution
When we have the fraction of the type , then we rationalize the denominator by multiplying and dividing the numerator by .
= 
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Example 12
Rationalize the following fraction:
Solution
When we get the fraction of the type , then to rationalize, we multiply the numerator and denominator by . In the above example, we have to rationalize the fraction , hence we will multiply the numerator and denominator by 
like this:
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= 
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= 
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