Rationalize Denominator Calculator
Rationalize a denominator — rewrite a fraction with a radical (square root, cube root, or higher) on the bottom so the denominator becomes a whole number, without changing the fraction's value. Handle a single radical term, a binomial with one radical, a binomial with two different radicals, or a higher-order root, each with full step-by-step reasoning and a visual showing exactly why the trick works.
Background
Rationalizing a denominator means multiplying a fraction by a clever form of 1 — something like √c/√c or a conjugate over itself — so that the radical disappears from the bottom and lands in the top instead. This never changes what the fraction equals, since multiplying by 1 can't change a value; it only changes how the value is written. For binomial denominators like p + q√r, the key tool is the conjugate, p − q√r, which uses the difference-of-squares pattern to erase the radical entirely.
How to use this calculator
- Choose Single Radical when the denominator is just a coefficient times one root, like b√c.
- Choose Binomial + Radical when the denominator has a whole-number part plus one radical term, like p ± q√r — this uses the conjugate trick.
- Choose Two-Radical Binomial when the denominator has two different radical terms, like √a ± √b — also solved with a conjugate.
- Choose Higher-Order Root for cube roots, fourth roots, or fifth roots in the denominator, like b · ⁿ√m.
- Click Calculate to see the visual breakdown, the full step-by-step algebra, and a callout explaining what the rationalized result actually means — including a decimal check confirming nothing changed.
How rationalizing a denominator works
Rationalizing a denominator means rewriting a fraction so no radical remains on the bottom, by multiplying the whole fraction by a form of 1 that clears the root.
For a single radical like b√c, multiplying top and bottom by √c works because √c × √c = c, a whole number.
For a binomial like p + q√r, the trick is the conjugate p − q√r: multiplying these together uses difference of squares, (p+q√r)(p−q√r) = p² − q²r, which eliminates the radical entirely.
The same conjugate idea works when both terms are radicals, like √a + √b: its conjugate is √a − √b, and (√a+√b)(√a−√b) = a − b.
Cube roots and higher roots need a different multiplier: to clear ⁿ√m you multiply by ⁿ√(m^(n−1)), because ⁿ√m × ⁿ√(m^(n−1)) = ⁿ√(m^n) = m.
Rationalizing never changes the value of the expression, only its form, since you're always multiplying by a fraction that equals exactly 1.
Formulas & Equations Used
Single radical: N / (b√c) = N√c / (bc)
Binomial + radical (conjugate): N / (p + q√r) = N(p − q√r) / (p² − q²r)
Two-radical binomial (conjugate): N / (√a + √b) = N(√a − √b) / (a − b)
Higher-order root: N / (b · ⁿ√m) = N · ⁿ√(m^(n−1)) / (bm)
Example Problems & Step-by-Step Solutions
Example 1 — A single radical
Rationalize 1/√2.
Step: Multiply top and bottom by √2: (1×√2)/(√2×√2) = √2/2.
Result: √2/2 — the same value, just without a radical on the bottom.
Example 2 — Simplify first, then rationalize
Rationalize 5/(2√8).
Step: √8 = 2√2, so the fraction is 5/(4√2). Multiply by √2/√2: 5√2/8.
Result: 5√2/8.
Example 3 — A binomial with one radical
Rationalize 1/(1+√2).
Step: Multiply by the conjugate (1−√2)/(1−√2). Denominator: 1−2 = −1. Numerator: 1−√2.
Result: (1−√2)/(−1) = √2 − 1.
Example 4 — Two different radicals
Rationalize 1/(√3+√2).
Step: Multiply by the conjugate (√3−√2)/(√3−√2). Denominator: 3−2 = 1.
Result: √3 − √2.
Example 5 — A cube root denominator
Rationalize 1/∛4.
Step: Multiply by ∛16/∛16, since ∛4×∛16 = ∛64 = 4. ∛16 simplifies to 2∛2.
Result: 2∛2/4 = ∛2/2.
Frequently Asked Questions
Why rationalize the denominator at all?
It's a convention that makes results easier to compare, add, and simplify — a radical-free denominator is easier to work with by hand, and it's the standard form expected on most tests and textbooks. The value of the fraction never changes; only its written form does.
What is a conjugate, and why does multiplying by it remove the radical?
The conjugate of p + q√r is p − q√r (same terms, opposite sign in the middle). Multiplying a sum by its conjugate always triggers the difference-of-squares pattern, (x+y)(x−y) = x² − y², which squares away the radical since (q√r)² = q²r is a whole number.
Does rationalizing change the value of the fraction?
No. Every rationalizing step multiplies by a fraction that equals exactly 1 (like √c/√c or a conjugate over itself), so the value stays identical — only its form changes. This calculator shows a decimal check confirming the before and after values match.
What if the denominator already has no radical?
Then there's nothing to rationalize — the fraction is already in simplest form with respect to radicals. This calculator assumes you're entering a denominator that does contain at least one radical term.
Why do cube roots need a different rationalizing factor than square roots?
A square root needs one more copy of itself to become whole (√c × √c = c), but a cube root needs two more copies (∛m × ∛m × ∛m = m), which is why the multiplier is ∛(m²) instead of ∛m. In general, an nth root needs n−1 extra copies, giving the multiplier ⁿ√(m^(n−1)).
What happens if two radicals combine into one after rationalizing?
Sometimes the numerator ends up with two radical terms that are secretly the same radical once simplified (for example, 2√2 and √2 are both multiples of √2). When that happens, this calculator combines them into a single term automatically, giving a cleaner final answer.
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