# Rationalize Denominator

How to rationalize the denominator (monomial, binomial radical): 4 examples and their solutions.

## Example√7y/√x

The denominator √x is a radical.

To rationalize the denominator √x,

multiply [√x/√x].

√7y⋅√x = √7xy

√x⋅√x = x

Multiply Radicals

So

√7xy/x

is the answer.

## Example√0.2

0.2 = 2/10

2/10 = 1/5

√1/5

= √1/√5

= 1/√5

Divide Radicals

The denominator √5 is a radical.

To rationalize the denominator √5,

multiply [√5/√5].

1⋅√5 = √5

√5⋅√5 = 5

So

√5/5

is the answer.

## Example1/[4 + √3]

The denominator 4 + √3

is a binomial radical.

To rationalize the denominator,

multiply, the conjugate of 4 + √3, 4 - √3

to both of the numerator and the denominator.

(The conjugate of [a + b] is [a - b].

The conjugate of [a - b] is [a + b].)

1⋅(4 - √3)

= 4 - √3

(4 + √3)(4 - √3)

= 4^{2} - (√3)^{2}

= 16 - 3

Product of a Sum and a Difference: (a + b)(a - b)

16 - 3 = 13

So

[4 - √3]/13

is the answer.

## Example√2/[5 - √6]

The denominator 5 - √6

is a binomial radical.

To rationalize the denominator,

multiply, the conjugate of 5 - √6, 5 + √6

to both of the numerator and the denominator.

√2(5 + √6)

= 5√2 + √2⋅√6

Common Monomial Factor

(5 - √6)(5 + √6)

= 5^{2} - (√6)^{2}

= 25 - 6

√6 = √2⋅√3

25 - 6 = 19

√2⋅√2 = 2

So

[5√2 + 2√3]/19

is the answer.