Consider the fraction , whose denominator is an irrational number.

Let us now multiply the numerator and denominator of this fraction by , obtaining an equivalent fraction:

Note that the equivalent fraction has a rational denominator.

We call this transformation **rationalization of denominators**.

The rationalization of denominators, therefore, consists in obtaining a fraction with rational denominator, equivalent to an earlier one, which had one or more radicals in its denominator.

To rationalize the denominator of a fraction, we must multiply the terms of this fraction by a radical expression, called a rationalizing factor, in order to obtain a new equivalent fraction with a denominator without radical.

## Main cases of rationalization

**1st case****:** **The denominator is an index 2 radical.** Example:

is the rationalizing factor of , because = The |

**2nd case****:** **The denominator is an index radical other than 2, or the sum (or difference) of two terms. **

In this case, it is necessary to multiply the numerator and denominator of the fraction by a convenient term so that the radical in the denominator disappears. Example:

The following are the main rationalizing factors according to the denominator type.

is the rationalizing factor of is the rationalizing factor of is the rationalizing factor of is the rationalizing factor of |

Here's another example:

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