How to Rationalize the Denominator (With Step-by-Step Examples)
- Tyler Buffone
- Oct 2
- 2 min read
Updated: Oct 9
What Does “Rationalize the Denominator” Mean?
To rationalize the denominator is to rewrite a fraction so the denominator has no radicals (no square roots). You do this by multiplying by a clever form of 1 that removes the radical from the bottom while keeping the value of the fraction the same.
Why teachers expect it:
It gives a standard “simplest form.”
It avoids radicals in the denominator, which makes later operations cleaner.
It shows algebraic control of radicals and conjugates.
The Big Idea
If the denominator has one term with a radical, multiply the top and bottom by the exact radical factor that clears it.
If the denominator has two terms, use the conjugate:
The conjugate of (a + b) is (a - b) and the conjugate of (a - b) is (a + b).
Throughout each procedure, you're multiplying by 1 in disguise, so the value does not change; you only change the form.
Example 1: Rationalizing the Denominator When It Has Only One Term
Simplify the following expression by rationalizing the denominator:
Goal: remove the square root of 6 from the denominator.
Steps:
Identify the radical factor in the denominator; in this case, it is the square root of 6.
Multiply the numerator and the denominator by the radical factor.
Simplify and reduce the fraction (if necessary).
See the following images that shows the implementation of the above steps:
Note: the fraction (3/24) reduces to (1/8).
Example 2: Rationalizing the Denominator When It Has Two Terms
Simplify the following expression by rationalizing the denominator:
Goal: remove radicals from the binomial denominator.
Steps:
Identify the conjugate of the denominator; it is simply the denominator with the minus sign swapped for a plus sign.
Multiply the numerator and the denominator by the conjugate.
Simplify and reduce the fraction (if necessary).
See the following images that shows the implementation of the above steps:
This is already in fully simplified form, so no further simplification is necessary.
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