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Rationalizing the Denominator
Rationalizing the denominator is a required step in simplifying a radical expression.
Think of it as being like putting a fraction in lowest terms.
How to rationalize
Steps
Is there an irrational root
(radical) in the denominator?
If yes, what can you multiply
the denominator by to eliminate
the radical?
Multiply the numerator and
denominator by the same thing
(just like with common
denominator).
Simplify where possible
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Examples
𝑦
2
5
4
√𝑥
Yes
√2
Yes
3
√3
Yes
√2 × √2 4√𝑥 × 4�𝑥 3
4
= √4
= �𝑥 4
=2
=𝑥
2
√2
×
√2
𝑦
√2 √𝑥
2 × √2
2
2√2
=
2
= √2
×
𝑦 √𝑥
𝑥
√𝑥
√𝑥
3
3
√3 × √9
3
= √27
=3
5
3
√9
×3
3
√3 √9
3
5√9
3
2
1
�
2
Yes
31⁄2
1 √1
� =
2 √2
31⁄2 × 31⁄2
= 31⁄2+1⁄2
=3
√2 × √2
=2
1 √1
� =
2 √2
√1
√2
√2
2
×
√2
√2
Yes,
31⁄2 = √3
31⁄2
×
31⁄2 31⁄2
2
2 × 31⁄2
3
2√3
=
3
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905.721.2000 ext. 2491
This document last updated: 6/25/2012
Student Academic Learning Services
Page 2 of 2
When the denominator is an irrational binomial
When you want to rationalize a denominator that has two things added or subtracted, you have
to multiply by the conjugate.
The conjugate is the same binomial with the sign in between reversed.
When you multiply a binomial by its conjugate, you always get a rational expression.
Examples:
Expression
√2 − √3
5 + √4
Conjugate
√2 + √3
5 − √4
Now a real example:
Simplify the expression:
Steps
�√2 − √3��√2 + √3� = 2 − 3 = −1
�5 + √4��5 − √4� = 25 − 4 = 21
√11
√2+√5
Multiply both the numerator and
denominator by the conjugate.
After you expand the bottom, the
two irrational terms cancel.
Simplify where possible.
Expression multiplied by conjugate
Results
√11
√2 + √5
×
√2 − √5
√2 − √5
=
=
√11�√2 − √5�
�√2 + √5��√2 − √5�
√11√2 − √11√5
2 + √5√2 − √5√2 − 5
√22 − √55
=
−3
Hooray! The denominator is rational now!!
You could also get rid of the negative in the denominator
by multiplying top and bottom by -1:
√22 − √55 −1
×
−3
−1
−√22 + √55
=
3
−
√55 √22
=
3
This is the final answer.
=
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 6/25/2012