# Rationalize Denominator Calculator

URL: https://speedycalculator.com/calculators/math/rationalize-denominator/
Last updated: 2026-10-30

> The rationalize denominator calculator removes square roots from the denominator of a/(b√c) or a/(b ± √c) by multiplying by √c or by the conjugate.

## Inputs

- Expression
- a
- b
- c (under the root)

## Results

- Rationalized form
- Decimal value
- Steps

## Formula

```
a/(b√c) = a√c / (bc)
a/(b + √c) = a(b − √c) / (b² − c)
```

## Explanation

A simplified answer traditionally has no radicals in the denominator. Rationalising makes expressions easier to compare and add, and was essential when calculations were done by hand with tables.

### How to use the Rationalize Denominator Calculator

- **Expression**
- **a**
- **b**
- **c (under the root)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

For a single radical the numerator and denominator are multiplied by √c. For a binomial denominator they are multiplied by its conjugate, using (b + √c)(b − √c) = b² − c. The result is simplified.

## Worked example

Inputs — Expression: a / (b + √c); a: 3; b: 2; c (under the root): 3.

Results — **Rationalized form = 6 − 3√3**; Decimal value = 0.8038475773; Steps = Multiply top and bottom by the conjugate 2 − √3.
The denominator becomes 2² − 3 = 1..

## Key terms

- **Conjugate**: The same binomial with the sign between the terms changed.

## FAQ

**Why rationalise a denominator?**
It is the conventional simplified form and makes sums of fractions easier.

**What is a conjugate?**
b − √c is the conjugate of b + √c; their product has no root.

**How do I rationalise 1/√2?**
Multiply by √2/√2 to get √2/2.

**Does rationalising change the value?**
No – you multiply by a form of 1.


## Sources

- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)
- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)