What this calculator does
A surd in the denominator is awkward to work with and, before calculators, awkward to evaluate by hand. Multiplying above and below by the conjugate clears it, because the difference of two squares turns the root into a whole number.
The value never changes, only the form. One over one plus root two becomes root two minus one, and both evaluate to 0.414214.
The formula
The fraction is multiplied by the conjugate of the denominator over itself. The denominator becomes b squared less c squared times n, which contains no root.
| Term | Meaning |
|---|---|
| Conjugate | The same expression with the sign between the terms reversed. |
| Difference of squares | Why the conjugate works: (b + c√n)(b − c√n) leaves no root. |
| Radicand | The number under the root sign. |
The inputs explained
| Field | What to enter |
|---|---|
| Numerator a | The numerator of the original fraction. |
| Denominator constant term b | The constant term in the denominator. |
| Denominator root coefficient c | The coefficient of the root in the denominator. |
| Radicand n | The number under the root sign. |
When to use it
Simplifying an algebraic expression
Standard form generally requires no radical in the denominator.
Preparing for further algebra
Expressions combine far more easily once denominators are rational.
Understanding why the value is unchanged
Multiplying by one in disguise changes the form and not the number.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the radicand change the result?
The same fraction with three different roots.
| Radicand n | Rationalized form | Original value |
|---|---|---|
| √2 | -1.000 + 1.000√2.00 | 0.414214 |
| √3 | -0.5000 + 0.5000√3.00 | 0.366025 |
| √5 | -0.2500 + 0.2500√5.00 | 0.309017 |
Questions
Why bother rationalising at all?
Historically because dividing by an irrational number by hand was far harder than multiplying by one. It survives as a convention because it makes expressions easier to compare and combine, since equivalent expressions then look the same.
Why does multiplying by the conjugate work?
Because it creates a difference of two squares. The cross terms cancel and the squared root becomes the radicand itself, leaving a denominator with no root in it.
What if b squared equals c squared times n?
The new denominator is zero, which means the original denominator was zero too. The expression is undefined, and the calculator reports that rather than dividing by zero.
Does this work for cube roots?
Not with this conjugate. Cube roots need a different multiplier based on the difference or sum of cubes factorisation, which involves three terms rather than two.
For splitting a fraction into simpler ones, see the partial fractions calculator. For solving quadratics that produce surds, see the quadratic calculator.