Rationalize the Denominator Calculator
Enter a fraction with a square root in the denominator below to instantly get the rationalized form.
For numerator / √radicand
How to Rationalize a Denominator
Multiply the numerator and denominator by the same radical that's in the denominator: a/√b = (a × √b) / (√b × √b) = (a√b) / b.
When the denominator is a binomial with a radical, like 1/(2 + √3), multiplying by the radical alone won't clear it. Instead, multiply by the conjugate — the same expression with the middle sign flipped — so (2 + √3)(2 − √3) = 4 − 3 = 1, using the difference of squares.
Worked example
Rationalize 5/√3.
5/√3 × √3/√3 = 5√3 / 3
Result: 5√3 / 3
A second example, with a binomial denominator: rationalize 1/(2 + √3).
Multiply by the conjugate: 1/(2 + √3) × (2 − √3)/(2 − √3) = (2 − √3) / (4 − 3)
Result: 2 − √3
Common mistake
A common mistake is multiplying only the denominator by the rationalizing factor and forgetting the numerator. Since you're really multiplying by a form of 1 (like √3/√3 or (2 − √3)/(2 − √3)), both the top and bottom must be multiplied, or the value of the fraction changes.
Keep going
- Rationalizing a denominator usually means multiplying by a square root; the square root calculator gives you that exact radical value on its own first. Square Root Calculator
Frequently Asked Questions
Why do you rationalize a denominator?
By convention, a fraction shouldn't have a radical (square root) left in the denominator. Rationalizing rewrites it in an equivalent form with the radical only in the numerator.
How do you rationalize a single-term radical denominator?
Multiply the top and bottom by that same radical. Since √b × √b = b, the denominator becomes a whole number: a/√b = (a√b)/b.
Does rationalizing change the value of the fraction?
No. Multiplying by √b/√b is really multiplying by 1, since the radical divided by itself equals 1 — so the value stays exactly the same, only the form changes.
How do you rationalize a denominator with two terms (a binomial)?
Multiply the numerator and denominator by the conjugate of the denominator — the same two terms with the sign between them flipped. That uses the difference-of-squares pattern, (a + b)(a − b) = a² − b², to eliminate the radical.
Why does multiplying by the conjugate get rid of the square root?
Because (a + √b)(a − √b) = a² − b, and squaring a square root cancels the radical entirely. The result is a whole number (or at least a radical-free expression) in the denominator.
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