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:

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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:

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## 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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## 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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## Example 6

Rationalize the following fraction:

### Solution

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## Example 7

Rationalize the following fraction:

### Solution

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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

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## Example 12

Rationalize the following fraction:

### Solution

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