Formula and Method for Rationalizing a Denominator
Rationalizing a denominator means rewriting a fraction so its denominator no longer contains a square root, while keeping the fraction's value exactly the same. This calculator works with fractions in the general form (a + b√n) / (c + d√n), which covers both a simple radical denominator (set b = 0, c = 0, d = 1, as in a/√n) and a binomial radical denominator (as in a/(c + d√n)). The tool multiplies the top and bottom by the conjugate of the denominator, c − d√n, and simplifies the result.
How the calculation works
Multiplying numerator and denominator by the conjugate (c − d√n) is the same as multiplying the whole fraction by 1, so the value never changes. On the bottom, the difference-of-squares identity (x + y)(x − y) = x² − y² applies with x = c and y = d√n, giving c² − d²(√n)² = c² − d²n — and since (√n)² = n exactly, the radical vanishes and the new denominator is the rational number R = c² − d²n. On top, expanding (a + b√n)(c − d√n) gives (ac − bdn) + (bc − ad)√n. The calculator then factors any perfect square out of n (so √n simplifies to k√m with m square-free), folds a rational √n into the whole-number part when n is a perfect square, and finally divides the numerator and denominator by their greatest common divisor to present the fraction in lowest terms.
Common mistakes
- Multiplying only the denominator: you must multiply both the numerator and the denominator by the same conjugate, or the fraction's value changes.
- Using the wrong conjugate: the conjugate of c + d√n is c − d√n (flip only the sign in front of the radical term, not the sign of c).
- Forgetting to simplify the radical first: if n has a perfect-square factor (like 8 = 4 × 2), pull it out — √8 = 2√2 — before or after rationalizing, so the final answer is in simplest form.
- Not reducing the final fraction: after clearing the radical, check whether the numerator terms and the new denominator share a common factor and divide it out.
Where this shows up
- Algebra and precalculus coursework, where a rational denominator is the required "simplified" form for a final answer.
- Evaluating limits in calculus, such as lim(h→0) (√(x+h) − √x)/h, which requires rationalizing the numerator or denominator before the limit can be taken directly.
- Simplifying exact-value results from the Pythagorean theorem, the distance formula, or the quadratic formula, which often produce a square root in a denominator.
- Standardized tests and textbooks, which conventionally expect radicals to be cleared from denominators in a final answer.