Rationalize Denominator Calculator

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.

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.

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.

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.

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.

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.

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.

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.