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Rationalize Denominator Calculator — result sheet
Remove one square-root term from a numeric denominator by multiplying by its conjugate, then show the equivalent radical form and a numeric check.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For a/(b+c√d), multiply numerator and denominator by b−c√d; the conjugate product is b²−c²d, so the denominator has no radical term.
A radical in a denominator can be removed without changing the value when the fraction is multiplied by a carefully chosen form of one. For a denominator made from a rational part and one square-root term, the conjugate changes the radical sign and makes the cross terms cancel.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Numerator — minimum -1000000000; maximum 1000000000
- Rational denominator part b — minimum -1000000000; maximum 1000000000
- Radical coefficient c — minimum -1000000000; maximum 1000000000
- Positive non-square radicand d — minimum 1; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Numerator | 5 |
| Rational denominator part b | 2 |
| Radical coefficient c | 1 |
| Positive non-square radicand d | 3 |
5/(2+√3) becomes 10 − 5√3; the numeric value is approximately 1.3397459622.
Assumptions and limits
- The denominator has the form b + c√d with one positive non-square integer radicand.
- The radical coefficient is nonzero so the page represents a genuine radical denominator.
- The original denominator and its conjugate product are nonzero.
- Inputs are numeric real values; symbolic variables and multiple unrelated radicals are outside the parser-free model.
- The returned coefficients are decimal representations of the equivalent rational and radical terms.
- A denominator that is already rational should be handled as an ordinary fraction rather than forced through this model.
- The numeric value check uses ordinary finite JavaScript arithmetic and can show rounding in extreme cases.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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