Rationalize the Denominator Calculator

About the Rationalize the Denominator Calculator

Rationalizing the denominator rewrites a fraction so that the denominator no longer contains radicals. It is a standard algebra skill because many textbooks, exams, and symbolic-manipulation rules expect expressions to be written in that form before further simplification.

This calculator covers four common cases. For monomial radicals like a/√c, you multiply by √c/√c. For binomial expressions like a/(b + √c), you multiply by the conjugate (b − √c) so the denominator becomes a difference of squares. It also handles expressions with two different radicals a/(√c − √d) and cube-root denominators a/∛c.

The before-and-after comparison shows how the numerator and denominator change, and the reference table puts the common rationalization patterns side by side so you can identify the right method for the form in front of you.

When This Page Helps

Use this page when you need to check the correct conjugate, confirm the transformed denominator, or compare the original and rationalized forms before simplifying further. It is most useful for algebra practice, tutoring, and test prep where the method matters as much as the final expression.

How to Use the Inputs

1. Enter Numerator (a) and Denominator rational part (b) in the input fields.
2. Select the mode, method, or precision options that match your rationalize the denominator problem.
3. Read Original Expression first, then use Rationalized Form to confirm your setup is correct.
4. Try a preset such as "1/(√2)" to test a known case quickly.

Example Calculation

Tips & Best Practices

• Always multiply by a form of 1 (conjugate/conjugate) so the expression value does not change.
• For binomial denominators, the conjugate flips the sign between the two terms.
• After rationalizing, simplify the resulting fraction by dividing numerator and denominator by their GCD.
• Cube roots require multiplying by ∛c² to get ∛c³ = c in the denominator.
• Check your answer by computing the decimal value — it should match the original expression.

Choosing the Right Rationalization Pattern

The denominator tells you which technique to use. A single square root usually needs the same radical above and below. A binomial with a radical usually needs the conjugate. Cube roots require multiplying by enough of the same root to build a perfect cube in the denominator.

Reading the Output

Start with the original expression, then look at the multiplier used to preserve the value of the fraction. After that, check the rationalized form and the decimal comparison to confirm that the transformed expression is equivalent to the original.

Why the Conjugate Works

For binomials such as b + √c, multiplying by b - √c creates (b + √c)(b - √c) = b² - c. The radical disappears because the middle terms cancel. That same pattern appears across many algebra and precalculus problems, so recognizing it here helps with later symbolic manipulation.

Frequently Asked Questions

• Why do we rationalize denominators?

  Convention in mathematics requires denominators to be rational. It also makes expressions easier to compare, add, and simplify further.

• What is a conjugate?

  The conjugate of (a + √b) is (a − √b). Multiplying an expression by its conjugate creates a difference of squares, eliminating the radical.

• Does rationalizing change the value of the expression?

  No — you multiply by a form of 1 (e.g., √c/√c = 1), so the value stays the same while the form changes.

• How do I rationalize a cube root in the denominator?

  Multiply numerator and denominator by ∛c², so the denominator becomes ∛c³ = c, a rational number.

• What if the denominator has two different square roots?

  Use the two-radicals mode. The conjugate of (√a − √b) is (√a + √b), giving denominator a − b.

• Can I rationalize denominators with higher-index roots?

  Yes — for an nth root, multiply by the (n−1)th power of the radicand under the root. This calculator covers square and cube roots, the most common cases.