Rationalize the Denominator Calculator — Remove √ from Denominator
Enter the numerator a and the radicand b (the integer under the square root in the denominator) to rationalize a/√b. The calculator extracts any perfect-square factor from √b, multiplies through by the conjugate radical, and reduces the resulting fraction to simplest form — showing every step.
1 / √3 = √3 / 3
Original expression
Multiply numerator and denominator by √3
Rationalized result
- 1
Square root of b (denominator)
√3 = 1.732051This is the irrational number we need to remove from the denominator. - 2
Numerator ÷ √b
1 ÷ 1.732051 = 0.577350
How does this calculator work?
To rationalize a/√b: multiply top and bottom by √b to get a√b/b. First simplify √b by extracting perfect-square factors (e.g. √12 = 2√3), then reduce the resulting fraction by the GCD. Example: 4/√6 → 4√6/6 → 2√6/3 (dividing by GCD(4, 6) = 2).
Formula
How this is calculated
Rationalizing the denominator means rewriting a fraction so that no square roots appear in the denominator. For a/√b, multiply both numerator and denominator by √b. The denominator becomes √b × √b = b (a rational integer), and the numerator becomes a√b. The result a√b/b has the radical only in the numerator, which is the conventional simplified form.
Before multiplying, it is efficient to simplify √b by extracting any perfect-square factor. For √12 = √(4 × 3) = 2√3, so 1/√12 = 1/(2√3), and multiplying by √3/√3 gives √3/6 rather than √12/12. Both are equal, but the form with the smaller square-free radicand is simpler. Once the multiplication is complete, the fraction coefficient a/(k × m) is reduced by the GCD of the numerator and denominator to reach lowest terms.
This calculator handles the case a/√b where a is any integer and b is any positive integer. More complex forms — such as a/(√b + √c) or a/(p + √b) — require multiplying by a conjugate (√b − √c or p − √b) to eliminate the radical using the difference-of-squares identity; those forms are not covered here.
Frequently asked questions
Historical convention and arithmetic practicality: it is easier to divide by a whole number than by an irrational one. In algebra it also makes it easier to add fractions and compare expressions, because two rationalized fractions share a rational denominator structure.
If b is a perfect square (4, 9, 16, …), then √b is an integer, so the original expression a/√b is already rational. The calculator will simplify it to a whole number or simple fraction with no radical in the result.
Multiply numerator and denominator by the conjugate (√b − √c). The denominator becomes (√b + √c)(√b − √c) = b − c using the difference-of-squares identity, eliminating both radicals. This calculator handles the simpler a/√b form only.
Also known as
TG we-Calculate Editorial Team. (2026). Rationalize the Denominator Calculator — Remove √ from Denominator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rationalize-denominator-calculator
TG we-Calculate Editorial Team. "Rationalize the Denominator Calculator — Remove √ from Denominator." TG we-Calculate. 2026. https://we-calculate.com/calculator/rationalize-denominator-calculator.
TG we-Calculate Editorial Team, "Rationalize the Denominator Calculator — Remove √ from Denominator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rationalize-denominator-calculator
@misc{wecalculate_rationalize_denominator_calculator, title = {Rationalize the Denominator Calculator — Remove √ from Denominator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rationalize-denominator-calculator}}, year = {2026}, note = {TG we-Calculate} }
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