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# Rationalising the denominator - easier questions
- URL: https://www.esheets.io/rationalising-the-denominator-easier-questions/
- Published: 2025-03-23T15:43:07.000Z
- Updated: 2026-06-21T16:47:04.000Z
- 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 simple surd in the denominator.
- Multiplying numerator and denominator by the same surd.
- 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 the denominator". In mathematics, it is considered bad practice to leave a surd (a square root) on the bottom of a fraction. Rationalising is the process of removing the root from the bottom, without changing the actual value of the fraction.

### Key method

The core technique relies on the rule that multiplying a square root by itself makes it a normal whole number (e.g. √3 × √3 = 3).

- Look at the fraction. Identify the surd on the bottom (e.g. √5).
- Multiply both the **top** and the **bottom** of the fraction by that exact same surd. You must do it to both top and bottom to keep the fraction equivalent.
- Multiply out the numerator (the top).
- Multiply out the denominator (the bottom). The roots will cancel each other out, leaving a normal integer.
- **Simplify:** Look at your final fraction. Can the outside numbers be simplified like a normal fraction?

### Worked example

**Rationalise the denominator of 6 / √2.**

Step 1: The surd on the bottom is √2\. We must multiply top and bottom by √2.

( 6 × √2 ) / ( √2 × √2 )

Step 2: Multiply the top.

6 × √2 = 6√2.

Step 3: Multiply the bottom.

√2 × √2 = √4 = 2.

Step 4: Combine the new fraction.

We now have 6√2 / 2.

Step 5: Simplify. The outside numbers are 6 and 2\. Because 6 divides by 2 exactly, we can simplify this.

6 ÷ 2 = 3.

The final, fully simplified answer is 3√2.

### Common mistakes to avoid

The most common mistake is forgetting to simplify the final fraction. If you leave the answer as 6√2 / 2, you will lose the final mark. Always check if the top whole number and the bottom whole number can be divided.

### Things to remember

You are not changing the size of the number, only its appearance. 6 / √2 and 3√2 are mathematically identical. You are essentially multiplying the fraction by 1 (because √2 / √2 is equal to 1), which is why the overall value doesn't change.