Rationalize the Denominator Calculator (square roots, cube roots & conjugates)

Understanding the rationalize the denominator calculator — concept and principle

This calculator rewrites a fraction so that no root remains in the denominator, without changing its value. It handles a single square root, a two-term denominator p + q√r and a single n-th root, simplifies the result and gives a decimal check. Use it for algebra homework and for writing answers in the form most textbooks require.

The difference of squares removes the root from the denominator without changing the value.

How it is calculated

One square root: a/(q√r) × √r/√r = a√r/(q·r).
Two terms: multiply top and bottom by the conjugate p − q√r, because (p + q√r)(p − q√r) = p² − q²r has no root.
n-th root: multiply by the root that completes a perfect n-th power, e.g. 1/∛2 × ∛4/∛4 = ∛4/2.

The root is simplified first (√12 = 2√3), and common factors of all the whole numbers are cancelled at the end. Whole numbers up to 1,000,000 are accepted.

The principle

Multiplying a fraction's top and bottom by the same non-zero number does not change its value. The trick is to choose a multiplier that turns the denominator into a whole number. For one square root, √r × √r = r. For a sum, the difference of squares (A + B)(A − B) = A² − B² squares the root away. For higher roots, the exponent inside the root is raised to a multiple of n. Historically this made hand division easier; today it gives a standard form that is easy to compare and simplify.

Worked example

Default inputs: a = 6, denominator 3 + √5 (p = 3, q = 1, r = 5).

Multiply by the conjugate 3 − √5: 6(3 − √5) ÷ ((3 + √5)(3 − √5)).
Denominator: 3² − 1² × 5 = 9 − 5 = 4.
Numerator: 18 − 6√5, so the fraction is (18 − 6√5)/4.
Cancel the common factor 2: (9 − 3√5) / 2.
Decimal check: (9 − 3 × 2.2360680) ÷ 2 ≈ 1.145898034, the same as 6 ÷ (3 + 2.2360680).

How to read the result

The rationalized form is equal to the original fraction; the decimal value confirms it. A whole-number denominator makes it easy to split the fraction into parts or to add it to other fractions. If the root in the numerator can still be simplified, the calculator has already done that.

Common mistakes

Multiplying only the denominator by the conjugate.
Using p + q√r again instead of the conjugate p − q√r.
Forgetting to cancel a common factor at the end.
For cube roots, multiplying by the same cube root once, which gives ∛4 in the denominator, not a whole number.

Keep in mind: The tool accepts whole-number inputs only. Denominators with variables or with two different roots, such as √2 + √3, need the same idea applied by hand.

Frequently asked questions

Why do teachers ask to rationalize the denominator?

It gives one standard form for answers and makes them easier to compare, add and estimate.

Does rationalizing change the value?

No. You multiply by a form of 1, such as √5/√5.

What is a conjugate?

For p + q√r, it is p − q√r: the same terms with the sign between them flipped.

Sources: OpenStax, Intermediate Algebra 2e (2020), §8.5 Divide Radical Expressions

Checked 9 Nov 2026. OpenStax, Intermediate Algebra 2e (2020), §8.5 Divide Radical Expressions — “Rationalize a One Term Denominator” (multiply numerator and denominator by the root that makes the radicand a perfect power of the index; simplify the radical first) and “Rationalize a Two Term Denominator” (multiply by the conjugate; the product of conjugates has no square root).

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