Use this calculator to rationalise expressions with radicals in the denominator. It supports both classic forms: a/√n and a/(b + c√n), and shows clear algebraic steps.
What does “rationalising the denominator” mean?
Rationalising the denominator means rewriting a fraction so that there are no radicals (like square roots) in the denominator. The value of the expression stays exactly the same; only the form changes.
This is a standard algebra technique because rational denominators are often easier to compare, simplify, and use in later steps (especially in equations, calculus, and exam settings).
Core formulas used by the calculator
1) Simple denominator
You multiply top and bottom by √n. The denominator becomes n because √n × √n = n.
2) Binomial denominator
Here you multiply by the conjugate of the denominator: (b - c√n). This works because:
Why this method works
Multiplying a fraction by a form of 1 does not change its value. For example, multiplying by √n/√n or by (b - c√n)/(b - c√n) preserves the original expression while removing radicals from the denominator.
- You preserve equality (same value).
- You get a cleaner denominator.
- You create a form that is easier to simplify.
Common mistakes to avoid
- Forgetting to multiply the numerator by the same factor as the denominator.
- Using the wrong conjugate for binomials.
- Sign errors when expanding terms like (b + c√n)(b - c√n).
- Not simplifying at the end if common factors exist.
Example walkthroughs
Example A: 7/√2
Multiply by √2/√2 to get (7√2)/2.
Example B: 4/(3 + √5)
Multiply by (3 - √5)/(3 - √5):
When is rationalisation useful?
Rationalisation appears in algebra, trigonometry, calculus limits, and exact-value geometry problems. Even when calculators can produce decimals, exact radical forms are often required in coursework and proofs.