Surds and rationalising the denominator

Q: How do you rationalise a denominator that contains two terms, like \(a - \sqrt{b}\)?

Multiply the top and bottom of the fraction by the conjugate , which is the same expression but with the opposite sign (e.g., \(a + \sqrt{b}\)).

Further Pure Mathematics · Pearson Edexcel IGCSE · 4 min read · Updated: Sep 7, 2026

Introduction to Surds

Welcome to the world of Surds! While you have worked with square roots before, Further Pure Mathematics requires a higher level of precision. Sometimes, writing a number as a decimal (like \(1.414...\)) just isn't accurate enough. Surds allow us to keep our answers exact.

In this chapter, we will learn how to "tidy up" these roots through simplification and rationalising the denominator. This topic is a foundational part of the Logarithmic functions and indices section of your Pearson Edexcel IGCSE (4PM1) course.

Note: For more on how surds relate to powers, see the chapter on "Indices and the functions \(a^x\)".

What is a Surd?

A surd is an irrational number that is expressed as a root (usually a square root). For example, \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) are surds because they cannot be written as simple fractions or terminating/recurring decimals. However, \(\sqrt{4}\) is not a surd because it equals exactly \(2\).

Key Rules to Remember

To manipulate surds, you need to be comfortable with these two fundamental rules:

1. Multiplication Rule: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)

2. Division Rule: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

Simple Manipulation of Surds

Don't worry if surds look intimidating! Think of them like algebraic variables (like \(x\) or \(y\)). You can only add or subtract them if they are "like terms."

Adding and Subtracting

You can add surds only if the number under the square root is the same.

Example: \(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\)

Imagine this as \(5x + 2x = 7x\). If the numbers inside the roots were different, like \(\sqrt{3} + \sqrt{2}\), you could not simplify them further.

Simplifying a Surd

To simplify a surd like \(\sqrt{48}\), we look for the largest square number that goes into the number under the root.

Step-by-step:
1. Find the factors of \(48\). Square numbers are \(4, 9, 16, 25, 36...\)
2. We see that \(16\) is a factor of \(48\) (\(16 \times 3 = 48\)).
3. Rewrite the surd: \(\sqrt{48} = \sqrt{16 \times 3}\)
4. Use the multiplication rule: \(\sqrt{16} \times \sqrt{3}\)
5. Since \(\sqrt{16} = 4\), the answer is \(4\sqrt{3}\).

Quick Review Box: Always check if your final surd can be simplified further. If you used \(\sqrt{4 \times 12}\) instead of \(16\), you would get \(2\sqrt{12}\). Since \(12\) still contains the square number \(4\), you would need to simplify again!

Rationalising the Denominator

In mathematics, it is considered "untidy" to leave a surd on the bottom of a fraction. Rationalising is the process of moving the surd to the top (the numerator) so that the bottom (the denominator) is a rational number (a whole number or a simple fraction).

Type 1: A Single Surd on the Bottom

If you have a fraction like \(\frac{10}{\sqrt{5}}\), you multiply both the top and the bottom by the surd that is in the denominator.

Step-by-step:
1. Multiply top and bottom by \(\sqrt{5}\):
\(\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\)
2. On the top: \(10 \times \sqrt{5} = 10\sqrt{5}\)
3. On the bottom: \(\sqrt{5} \times \sqrt{5} = 5\) (Remember: \(\sqrt{a} \times \sqrt{a} = a\))
4. The fraction is now \(\frac{10\sqrt{5}}{5}\)
5. Simplify the whole numbers: \(10 \div 5 = 2\)
Final Answer: \(2\sqrt{5}\)

Type 2: A Denominator with Two Terms

When the denominator looks like \(a + \sqrt{b}\) or \(a - \sqrt{b}\), we use a trick called the conjugate. This involves multiplying by the same expression but with the opposite sign.

Example: Rationalise \(\frac{1}{2 - \sqrt{3}}\)

Step-by-step:
1. The denominator is \(2 - \sqrt{3}\). The conjugate is \(2 + \sqrt{3}\).
2. Multiply top and bottom by the conjugate:
\(\frac{1}{(2 - \sqrt{3})} \times \frac{(2 + \sqrt{3})}{(2 + \sqrt{3})}\)
3. The top becomes: \(1 \times (2 + \sqrt{3}) = 2 + \sqrt{3}\)
4. The bottom becomes: \((2 - \sqrt{3})(2 + \sqrt{3})\)
5. Expand the bottom (using FOIL): \(4 + 2\sqrt{3} - 2\sqrt{3} - 3\)
6. The middle terms cancel out! The bottom is just \(4 - 3 = 1\).
Final Answer: \(\frac{2 + \sqrt{3}}{1} = 2 + \sqrt{3}\)

Did you know? This trick works because of the "difference of two squares" identity: \((x - y)(x + y) = x^2 - y^2\). By squaring both parts, we get rid of the root!

Common Mistakes to Avoid

1. Adding different roots: \(\sqrt{2} + \sqrt{3}\) is NOT \(\sqrt{5}\). You cannot combine them unless the number inside the root is the same.

2. Squaring incorrectly: \((\sqrt{5})^2\) is \(5\). Many students accidentally write \(25\).

3. Sign errors: When rationalising something like \(\frac{1}{5 + \sqrt{2}}\), you must multiply by \(5 - \sqrt{2}\). If you use the same sign, the surds on the bottom won't cancel out.

Key Takeaways

Surds are exact values for irrational roots.
Simplify by finding the largest square factor (e.g., \(\sqrt{48} = 4\sqrt{3}\)).
Add/Subtract only "like" surds (e.g., \(5\sqrt{3} + 2\sqrt{3}\)).
Rationalise by multiplying the top and bottom to remove the root from the denominator.
For complex denominators, use the conjugate (opposite sign).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a surd and why do we use them in Further Pure Mathematics?

A surd is an irrational number expressed as a root (e.g., \(\sqrt{2}\)) that cannot be written as a simple fraction or terminating decimal; we use them to maintain exact precision.

What are the fundamental rules for multiplying and dividing surds?

The rules are \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\) for multiplication and \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) for division.

Under what condition can you add or subtract two surds?

You can only add or subtract surds if they are "like terms," meaning the number under the square root is exactly the same.

How do you simplify a surd like \(\sqrt{48}\)?

Find the largest square number that is a factor of the value under the root, rewrite the surd as a product, and then take the square root of that factor.

What does it mean to "rationalise the denominator"?

Rationalising the denominator is the process of removing a surd from the bottom of a fraction so that the denominator becomes a rational number.

How do you rationalise a fraction with a single surd denominator, such as \(\frac{10}{\sqrt{5}}\)?

Multiply both the numerator and the denominator by that same surd.

How do you rationalise a denominator that contains two terms, like \(a - \sqrt{b}\)?

Multiply the top and bottom of the fraction by the conjugate, which is the same expression but with the opposite sign (e.g., \(a + \sqrt{b}\)).

Is \(\sqrt{2} + \sqrt{3} = \sqrt{5}\)?

No, you cannot add the numbers inside the roots together; they must remain as separate terms unless the roots are identical.

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