Rationalize Denominator Calculator

Rationalize Denominator Calculator. Enter a fraction with a radical in the denominator — choose your denominator type (simple square root, higher-order root, or binomial), fill in the numerator and denominator values, and get back the rationalized form with the multiplication step shown. Supports square roots, cube roots, and the conjugate method for binomial denominators. Also try the Unit Rate Calculator.

Choose the structure of your denominator.

Enter the number on top of the fraction.

The number inside the square root or nth root in the denominator.

Only used when denominator type is Higher-Order Root.

The integer term in the binomial denominator.

Results

Rationalized Numerator

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Rationalized Denominator

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Decimal Value of Original Fraction

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Decimal Value After Rationalization

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Method Used

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Results Table

Why do we rationalize the denominator?

Rationalizing the denominator is a standard mathematical convention that removes irrational numbers (like square roots) from the bottom of a fraction. It makes expressions easier to compare, simplify further, and work with in equations — and is typically required for a fully simplified answer in algebra and calculus. See also our Mixed Number Calculator.

What does it mean for a denominator to be irrational?

An irrational denominator contains a radical (such as √2 or ∛5) that cannot be expressed as a simple fraction. For example, 1/√3 has an irrational denominator. Rationalizing converts it to an equivalent fraction with a rational (integer or simple fraction) denominator, like √3/3.

How do you rationalize a simple square root in the denominator?

Multiply both the numerator and denominator by the same square root in the denominator. For example, to rationalize 3/√5, multiply top and bottom by √5 to get 3√5/5. The denominator becomes √5 × √5 = 5, which is rational.

What is the conjugate method for binomial denominators?

When the denominator is a binomial like (2 + √5), you multiply both numerator and denominator by its conjugate (2 − √5). This uses the difference of squares identity: (a+b)(a−b) = a² − b², which eliminates the radical from the denominator.

How do you rationalize a cube root or higher-order root in the denominator?

For an nth root, you multiply numerator and denominator by the (n−1)th power of the radical. For example, to rationalize 1/∛5, multiply by ∛(5²) = ∛25 to get ∛25/5, because ∛5 × ∛25 = ∛125 = 5.

Does rationalizing the denominator change the value of the fraction?

No. Rationalization multiplies both numerator and denominator by the same expression (effectively multiplying by 1), so the value of the fraction stays exactly the same. Only its written form changes.

Can a fraction have both an irrational numerator and denominator?

Yes — for example, √2/√3. After rationalizing the denominator (multiplying by √3/√3), you get √6/3. The numerator now contains a radical, but that is acceptable since only the denominator needs to be rational for a fully simplified form.

What are common mistakes when rationalizing denominators?

Common errors include forgetting to multiply the numerator when rationalizing, using the wrong conjugate sign (using (a+b) instead of (a−b)), and not simplifying the resulting expression fully. With higher-order roots, forgetting to raise the radical to the correct power (n−1) is also a frequent mistake. You might also find our calculate Egyptian Fractions Number of Unit Fractions useful.