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# Rationalising the denominator - medium difficulty
- URL: https://www.esheets.io/rationalising-the-denominator-medium-difficulty/
- Published: 2025-03-23T17:54:49.000Z
- Updated: 2026-06-21T16:47:05.000Z
- Author: Richard Linnington
- Tags: Maths, Number

Rationalising the denominator is a useful skill that often pops up when working with fractions in algebra and trigonometry. It's especially helpful in simplifying expressions, making them easier to work with in real-life applications like physics and engineering, where messy square roots can complicate calculations. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise rationalising the denominator with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Identifying a surd in the denominator.
- Multiplying numerator and denominator by a suitable surd or conjugate.
- Simplifying the denominator.
- Writing the answer without a surd in the denominator.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Topic guide

### What this worksheet practises

This worksheet provides practice on rationalising denominators where the bottom of the fraction contains a number *and* a root multiplied together (e.g. 5√3). The goal is to remove the root from the bottom without changing the value of the fraction.

### Key method

You only need to multiply by the root part, not the whole bottom expression.

- Identify the surd part of the denominator (e.g. in 4√7, the surd is just √7).
- Multiply both the **top** and the **bottom** of the fraction by this surd. Do not multiply by the whole 4√7.
- Multiply the numerator (the top). If the top has multiple terms, remember to multiply all of them by the surd.
- Multiply the denominator (the bottom). The roots will combine to form a whole integer, which is then multiplied by the integer already there.
- Simplify the final fraction if the outside numbers share a common factor.

### Worked example

**Rationalise the denominator of 10 / 3√5.**

Step 1: The surd part is √5\. We multiply top and bottom by √5.

( 10 × √5 ) / ( 3√5 × √5 )

Step 2: Multiply the top.

10 × √5 = 10√5.

Step 3: Multiply the bottom. (√5 × √5 = 5).

3 × 5 = 15.

Step 4: Combine the new fraction.

We now have 10√5 / 15.

Step 5: Simplify. The outside numbers (10 and 15) both divide by 5.

The fully simplified answer is 2√5 / 3.

### Common mistakes to avoid

A common mistake is multiplying top and bottom by the *entire* denominator (e.g. multiplying by 3√5). While this will eventually give you the correct answer, it creates unnecessarily large numbers (30√5 / 45) that are much harder to simplify at the end. Only multiply by the root.

### Things to remember

If the numerator is a complex expression like (2 + √3) and you are rationalising by multiplying by √5, you must treat the top like a bracket: √5 × (2 + √3) = 2√5 + √15\. Every part of the top gets multiplied.