Rationalize Denominator

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

Example 1

Example

Solution

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 2

Example

Solution

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.

Example 3

Example

Solution

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)
= 42 - (√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 4

Example

Solution

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)
= 52 - (√6)2
= 25 - 6

6 = √2⋅√3

25 - 6 = 19

2⋅√2 = 2

So [5√2 + 2√3]/19 is the answer.