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# Rationalising the denominator - harder problems
- URL: https://www.esheets.io/rationalising-the-denominator-harder-problems/
- Published: 2025-03-23T19:11:15.000Z
- 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 harder problems 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 binomial denominator involving a surd.
- Multiplying by the conjugate.
- Using the difference of two squares where relevant.
- Simplifying the final expression 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 focuses on the hardest type of rationalising questions, where the denominator (bottom) of the fraction contains two terms (e.g. 3 + √2). You cannot simply multiply top and bottom by √2, as this will not remove the root from the bottom.

### Key method

You must multiply the top and the bottom by the "conjugate" of the denominator. The conjugate is the exact same expression, but with the **opposite sign** in the middle.

- Identify the denominator (e.g. 5 − √3). Its conjugate is 5 + √3.
- Multiply the top of the fraction by this conjugate. This often requires expanding double brackets.
- Multiply the bottom of the fraction by this conjugate. If done correctly, this will always create a "difference of two squares" expansion. The middle terms will cancel out, and the roots will vanish, leaving an integer.
- Write out the new fraction and simplify it if possible.

### Worked example

**Rationalise the denominator of 4 / (3 + √2).**

Step 1: Find the conjugate of the bottom. It is (3 − √2).

Step 2: Multiply the top by the conjugate.

4 × (3 − √2) = 12 − 4√2.

Step 3: Multiply the bottom by the conjugate (expanding double brackets).

(3 + √2)(3 − √2) = 9 − 3√2 + 3√2 − 2.

The middle roots cancel out (+3√2 and −3√2), leaving 9 − 2 = 7.

Step 4: Combine the new fraction.

The final answer is (12 − 4√2) / 7.

### Common mistakes to avoid

The most common error is miscalculating the final term when expanding the bottom brackets. In our example, students correctly calculate √2 × −√2, but mistakenly write down −4 instead of −2\. Remember that a root times itself just removes the root symbol.

### Things to remember

When multiplying the bottom (a + √b)(a − √b), the shortcut is simply a² − b. This works every single time and saves you from writing out the full double bracket expansion. In our example, 3² − 2 = 9 − 2 = 7.