Rationalize Denominator Calculator
Remove square roots from the denominator of a fraction, simplify radicals, and see each algebra step clearly.
Radical denominator solver
A rationalize denominator calculator rewrites a fraction so the denominator no longer contains a square root. For a single radical denominator, it multiplies by the same radical. For a binomial denominator, it multiplies by the conjugate.
A rationalize denominator calculator removes radicals or irrational numbers from the denominator of a fraction. The calculator multiplies the numerator and denominator by a radical expression that eliminates the square root or cube root. For example, 1 / sqrt(2) becomes sqrt(2) / 2 after rationalizing the denominator.
Use this calculator for expressions like 5 / sqrt(3), 7 / (2sqrt(5)), or 4 / (3 + sqrt(2)). The result keeps exact radical form, reduces fractions when possible, and also shows a decimal approximation.
Rationalized result
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Original expression
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Input fraction
Multiplier
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Equivalent fraction used
Rational denominator
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No radical remains below the fraction bar
Decimal check
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Approximate value
Step-by-step rationalization
How to use the rationalize denominator calculator
- Choose the denominator type: Select single radical for expressions like n / sqrt(b), cube root for n / cbrt(b), or binomial for expressions like n / (a + b sqrt(c)).
- Enter the numerator: Type the integer on top of the fraction.
- Enter denominator values: Fill in the coefficient, rational term, and radicand fields that apply to your expression.
- Calculate: The tool multiplies by the correct radical or conjugate and simplifies the result.
- Check the steps: Use the step-by-step section to see why the denominator becomes rational.
Rationalizing denominator formulas
Rationalizing the denominator means rewriting an equivalent fraction so the denominator is a rational number. This is often required in algebra because it makes exact answers easier to compare, add, subtract, and simplify.
n / sqrt(b) × sqrt(b) / sqrt(b) = n sqrt(b) / b
n / (a + b sqrt(c)) × (a - b sqrt(c)) / (a - b sqrt(c))
(a + b sqrt(c))(a - b sqrt(c)) = a^2 - b^2 c
The conjugate method works because it uses the difference of squares pattern. The radical terms cancel in the denominator, leaving a rational denominator.
Reference: University of Houston - Rationalizing Denominators.
Which rationalizing method should you use?
The right method depends on the shape of the denominator. Before you multiply anything, identify whether the denominator is a single radical, a radical with a coefficient, or a two-term expression that needs a conjugate.
| Denominator shape | Example | Use this multiplier | Reason |
|---|---|---|---|
| Single square root | 1 / sqrt(7) | sqrt(7) / sqrt(7) | sqrt(7) times sqrt(7) equals 7. |
| Coefficient times radical | 5 / (3sqrt(2)) | sqrt(2) / sqrt(2) | Only the radical part needs another sqrt(2). |
| Rational plus radical | 6 / (4 + sqrt(3)) | (4 - sqrt(3)) / (4 - sqrt(3)) | The conjugate cancels the middle radical terms. |
| Rational minus radical | 2 / (5 - 2sqrt(6)) | (5 + 2sqrt(6)) / (5 + 2sqrt(6)) | The sign flips so the denominator becomes a difference of squares. |
Method reference: Portland Community College ORCCA - Rationalizing the Denominator.
Simplify before and after rationalizing
Many wrong answers come from rationalizing correctly but stopping before the expression is fully simplified. Use this checklist to keep the numerator, denominator, radical, and fraction reduction organized.
1. Simplify the radicand
Look for square factors first. For example, sqrt(20) becomes 2sqrt(5), which can make later fraction simplification easier.
2. Reduce common factors
After rationalizing, check whether the integer coefficient in the numerator and the rational denominator share a common factor.
3. Check equivalent value
Compare the decimal value of the original expression and the final answer. They should match, even though the exact form looks different.
What about cube roots and higher roots?
This calculator is built for square-root denominators, including binomial square-root denominators that require a conjugate. Cube roots and higher roots use a different idea: multiply by enough matching radical factors to make the exponent inside the denominator a full power.
1 / cbrt(2) needs cbrt(4) / cbrt(4)
cbrt(2) x cbrt(4) = cbrt(8) = 2
1 / cbrt(2) = cbrt(4) / 2
If your assignment includes cube roots, use the same goal: make the denominator rational without changing the value of the fraction. For most school algebra problems, square-root rationalization and conjugates are the main cases.
Higher-root reference: West Texas A&M University Virtual Math Lab - Radical Expressions.
Worked examples
| Expression | Multiplier | Rationalized form | Why it works |
|---|---|---|---|
| 5 / sqrt(3) | sqrt(3) / sqrt(3) | 5sqrt(3) / 3 | sqrt(3) times sqrt(3) equals 3. |
| 7 / (2sqrt(5)) | sqrt(5) / sqrt(5) | 7sqrt(5) / 10 | The denominator becomes 2 times 5. |
| 4 / (3 + sqrt(2)) | (3 - sqrt(2)) / (3 - sqrt(2)) | (12 - 4sqrt(2)) / 7 | The denominator becomes 3^2 - 2 = 7. |
| 1 / cbrt(2) | cbrt(4) / cbrt(4) | cbrt(4) / 2 | The denominator becomes cbrt(8), which equals 2. |
Common mistakes to avoid
The most common rationalizing mistakes happen when the numerator is changed without changing the denominator, or when the conjugate sign is chosen incorrectly.
Multiply by 1
Always multiply the numerator and denominator by the same expression. That keeps the fraction equivalent.
Use the conjugate
For a + sqrt(b), multiply by a - sqrt(b). For a - sqrt(b), multiply by a + sqrt(b).
Simplify radicals
If sqrt(12) appears, rewrite it as 2sqrt(3) when possible. This can reduce the final fraction.
Interesting fact
According to Wolfram MathWorld, sqrt(2) begins 1.4142135623.... That means 1 / sqrt(2) is about 0.7071067812, and sqrt(2) / 2 gives the same value with a rational denominator. This small example shows why rationalizing changes the form of an expression, not its value.
Frequently Asked Questions
What does it mean to rationalize denominator in algebra?
To rationalize denominator in algebra means to rewrite a fraction so the denominator contains no radical or irrational number. The value of the expression does not change because the numerator and denominator are multiplied by a form of 1, such as sqrt(3) / sqrt(3) or a conjugate over itself. The final answer is an equivalent expression with a rational number below the fraction bar.
When does the calculator use a conjugate?
The calculator uses a conjugate when the denominator is a binomial radical expression, such as 4 + sqrt(7) or 3 - 2sqrt(5). The conjugate has the same terms but the opposite sign between them, so multiplying creates a difference-of-squares equation in the denominator. That step removes the square root and produces a rational denominator.
Does rationalizing change the value of the expression?
No. Rationalizing changes the form of the fraction, not its value. The original expression and the rationalized expression should have the same decimal approximation, which is why this calculator shows a decimal check in the result panel. If the decimal result changes, there is usually a simplification or arithmetic step to review.
Can the rational denominator result be negative?
Yes. In a binomial case, the rational denominator is a^2 - b^2c. If b^2c is larger than a^2, the denominator result is negative. The fraction is still valid, and the negative sign can be kept in the denominator or moved to the numerator depending on the answer format your math class expects.
Can the calculator rationalize a cube-root denominator?
Yes. Choose cube-root mode for expressions like 1 / cbrt(2) or 5 / (3cbrt(4)). The calculator multiplies by the cube-root factor needed to complete a perfect cube in the denominator, then simplifies the numerator, denominator, and final fraction.
Why do teachers often prefer a rationalized answer?
A rationalized answer makes exact forms easier to compare, combine, and grade. If two students write equivalent solutions in different forms, a rationalized denominator can make it easier to see whether the results match. It also helps when a variable, radical, or square root appears later in a longer algebra simplification problem.
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Disclaimer: This rationalize denominator calculator is for general educational and informational use only. It is designed to support algebra practice and exact-form simplification. It may not match every classroom convention, teacher preference, textbook notation, or assessment format. Always review whether your course expects radicals to be simplified, fractions to be reduced, and negative signs to be placed in a specific location.
Last updated: May 19, 2026