# Rationalizing a Denominator

How to rationalize the radicals in a denominator: examples and their solutions.

## Example 1

Split the radical sign into two parts:

the numerator and the denominator.

Simplify the radicals

in the numerator and the denominator.

Simplifying a radical (part 2)

To rationalize √2 in the denominator,

multiply √2

to both of the numerator and the denominator.

Then √2⋅√2 = 2.

So the denominator is rationalized.

## Example 2

To rationalize √5,

multiply √5

to both of the numerator and the denominator.

## Example 3

To rationalize 4 + √3,

multiply its conjugate, 4 - √3,

to both of the numerator and the denominator.

Then (4 + √3)(4 - √3) = 4^{2} - (√3)^{2} = 13.

The denominator is rationalized.

Product of a sum and a difference

4 + √3 and 4 - √3

are the conjugates of each other.

Conjugates: *a* + *b* and *a* - *b*.

## Example 4

To rationalize 5 - √6,

multiply its conjugate, 5 + √6,

to both of the numerator and the denominator.

Numerator: √2(5 + √6) = 5√2 + 2√3

Multiplying radicals (part 2)

## Example 5

To rationalize √*x* - 2,

multiply its conjugate, √*x* + 2,

to both of the numerator and the denominator.

Cancel (*x* - 4).

Simplifying rational expressions