Rationalize Denominator Calculator
Rationalize square-root denominators with single-radical and conjugate methods, exact fraction coefficients, simplified roots, and step-by-step algebra.
Last Updated: May 2026
Use an integer, decimal, or fraction.
The k in k sqrt(n).
Positive whole number under sqrt.
Final answer
5/18 sqrt(3)
Original expression
5 / 6 sqrt(3)
Multiplier
sqrt(3) / sqrt(3)
New denominator
18
| Step | Value | Reason |
|---|---|---|
| Simplify radical | sqrt(12) = 2 sqrt(3) | Pull perfect-square factors outside first. |
| Multiply by | sqrt(3) / sqrt(3) | This equals 1, so the expression value is unchanged. |
| Denominator product | 6 sqrt(3) x sqrt(3) = 18 | sqrt(a) x sqrt(a) = a. |
| Final answer | 5/18 sqrt(3) | The denominator is rational. |
Multiply by the remaining square root so sqrt(n) times sqrt(n) becomes n.
For a + b sqrt(n), multiply by a - b sqrt(n) so the radical terms cancel.
Coefficients are kept as fractions so the final expression stays exact.
Rationalizing changes the form, not the value. The expression becomes easier to compare, combine, or use in algebra because the denominator no longer contains a radical.
Algebra Scope Notice
This calculator handles square-root denominator rationalization for single radical denominators and binomial conjugates. It is not a full symbolic algebra parser.
Checked by Jitendra Kumar
Rationalize Denominator Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
How to Use the Rationalize Denominator Calculator
Choose the single-radical mode for denominators like k sqrt(n), or choose the binomial conjugate mode for denominators like a + b sqrt(n). Enter coefficients as integers, decimals, or fractions.
The calculator simplifies the square root first, then shows the multiplier, the new numerator, the rationalized denominator, and the final equivalent expression.
Step 1: Choose the denominator type
Use single radical for k sqrt(n), or binomial conjugate for a + b sqrt(n).
Step 2: Enter the numerator
Use a whole number, decimal, or fraction such as 7 or 3/2.
Step 3: Enter denominator values
Fill in the coefficient, rational part, radical coefficient, and positive radicand as needed.
Step 4: Read the rationalized form
Use the final answer row when you need an equivalent expression with no radical in the denominator.
How This Rationalize Denominator Calculator Works
For a single radical denominator, the calculator first simplifies the square root. If a radical remains, it multiplies the fraction by that radical over itself so the denominator becomes rational.
For a binomial denominator, the calculator multiplies by the conjugate. The product follows the difference-of-squares pattern, so (a + b sqrt(c))(a - b sqrt(c)) becomes a squared minus b squared times c.
Coefficients are parsed as exact rational numbers. That keeps fraction answers exact and avoids decimal rounding in the algebra steps.
Rationalizing Denominators Guide
Rationalizing Rules
| Case | Example | Method |
|---|---|---|
| Single radical | 5 / sqrt(3) | Multiply by sqrt(3) / sqrt(3). |
| Coefficient radical | 5 / (3 sqrt(12)) | Simplify sqrt(12), then multiply by the remaining radical. |
| Binomial denominator | 7 / (2 + 3 sqrt(5)) | Multiply by the conjugate 2 - 3 sqrt(5). |
| Difference of squares | (a + b sqrt(c))(a - b sqrt(c)) | The denominator becomes a^2 - b^2c. |
| Perfect-square radicand | sqrt(16) | The denominator is already rational after simplification. |
Worked Examples
| Expression | Rationalized Form | Notes |
|---|---|---|
| 7 / sqrt(5) | 7 sqrt(5) / 5 | Single-radical rationalization. |
| 5 / (3 sqrt(12)) | 5 sqrt(3) / 18 | sqrt(12) simplifies to 2 sqrt(3). |
| 7 / (2 + 3 sqrt(5)) | (-14 + 21 sqrt(5)) / 41 | Conjugate method. |
| 4 / (6 - sqrt(11)) | 24/25 + 4 sqrt(11)/25 | Use the conjugate 6 + sqrt(11). |
| 3/2 / (5 sqrt(18)) | sqrt(2) / 20 | Exact fraction numerator is preserved. |
Why Conjugates Work
Conjugates work because the radical terms cancel in the middle. Multiplying a + b sqrt(c) by a - b sqrt(c) leaves a rational denominator, a squared minus b squared times c. The numerator may still contain a radical, which is acceptable because the denominator has been rationalized.
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- 1.Khan Academy - Rationalizing denominators(Accessed May 2026)
- 2.Wikipedia - Rationalisation(Accessed May 2026)
- 3.Wikipedia - Conjugate(Accessed May 2026)