How this calculator works
A fraction is fully simplified when no square root sits in the denominator. For a single radical the trick is to multiply the top and bottom by the same radical: 1/√b becomes √b/b, because √b × √b = b. For a binomial denominator such as a + √b, multiply by its conjugate a - √b. The product (a + √b)(a - √b) = a² - b contains no radical, so the denominator becomes a plain integer.
Formula
n/√b = n√b/b. For a binomial, n/(a + √b) = n(a - √b)/(a² - b). Reduce the resulting fraction by the greatest common factor of the numerator and denominator.
Worked example
Rationalize 3/√2: multiply by √2/√2 to get 3√2/2. Rationalize 1/(2 + √3): multiply by (2 - √3)/(2 - √3), giving (2 - √3)/(4 - 3) = 2 - √3.
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Frequently asked questions
Why rationalize a denominator?
A rational denominator is easier to compare and to add to other fractions, and it matches the standard form expected in textbooks. The value is unchanged; only the spelling changes.
What is a conjugate?
The conjugate of a + √b is a - √b. Multiplying the two gives a² - b, which has no radical because the cross terms cancel. The conjugate of a - √b is a + √b.
What happens when the radicand is a perfect square?
Then the square root is a whole number and the denominator already has no radical, so nothing needs rationalizing. This calculator asks for a different radicand in binomial form and simply simplifies in single-root form.
Can the numerator be zero?
Yes. A zero numerator makes the whole fraction zero, whatever the denominator. The rationalized form is reported as 0.