Rationalizing the Denominator
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
Rationalizing the denominator refers to the process of eliminating any square roots or irrational numbers from the denominator of a fraction. The goal is to rewrite the fraction in such a way that the denominator no longer contains a radical (such as a square root).
Why do we do it?
In mathematics, it is typically preferred to have a rational number (a number that can be expressed as a fraction of two integers) in the denominator, rather than a square root or other irrational number.
How to Rationalize the Denominator:
1. When the denominator is a square root: Multiply both the numerator and the denominator by the same square root to "cancel out" the radical in the denominator.
Example 1: Rationalizing a simple square root in the denominator:
1 / sqrt(2) denominator has a square root
Multiply both the numerator and the denominator by sqrt(2):
Now, the denominator is rational (just 2), and the fraction is rationalized.
2. When the denominator is a binomial with a square root: Multiply both the numerator and the denominator by the conjugate of the denominator (the same binomial, but with the opposite sign between terms).
Example 2: Rationalizing a denominator with a binomial:
Multiply both the numerator and denominator by sqrt(3) - 1 (the conjugate):
Now the denominator is rationalized.
Key Points:
Rationalizing the denominator eliminates square roots (or other radicals) from the denominator.
You can rationalize a denominator by multiplying both the numerator and denominator by the same radical or by using the conjugate when the denominator is a binomial.
This process makes the expression easier to work with and is generally preferred in mathematical conventions.