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Surds and Rationalising

Q: How do I simplify a surd like 72?

Find the largest square factor. 72 = 36 2, so 72 = 36 2 = 6 2. If you miss the largest factor, simplify again until no square factors remain.

A surd is a root that stays irrational, like $2$ . Keeping surds exact — instead of rounding to a decimal — is how A-Level answers stay precise, and rationalising the denominator is the standard way to write them tidily.

30 min Video by TLMaths Algebra and Functions

The big picture

You already proved in the Proof chapter that $2$ is irrational: its decimal never ends and never repeats, so $1.41$ is only ever an approximation. That is the whole reason surds exist as a topic. When a question says “give your answer in exact form”, it is telling you that rounding throws away information — the true value is a surd, and you must carry it through the algebra untouched. This lesson is about manipulating those exact values confidently: simplifying them, combining them, and removing them from denominators where convention (and easier arithmetic) demands.

A-Level Maths B2-01: Introducing Surds and Simplifying SurdsWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Simplify a surd by extracting square factors
Add, subtract and multiply surds correctly
Rationalise a denominator containing a single surd
Rationalise using the conjugate when the denominator is $a \pm b$

What a surd is (and why we keep it)

A $surd$ is a root of a whole number that cannot be simplified to a rational number — $2$ , $3$ , $5$ and so on. Roots like $9 = 3$ are $not$ surds, because they come out exactly.

Because a surd’s decimal expansion is infinite and non-repeating, writing $2 \approx 1.414$ loses accuracy immediately. Exam answers in “exact form” must therefore keep the surd, so that $2 \times 2$ stays the perfect $2$ rather than $1.41 4^{2} = 1.999 \dots$

Why this works

Working in surds is not fussiness — it is the only way to stay exact. Every rounding you avoid is an error you never make.

Simplifying surds

The key rule is that a root of a product splits into a product of roots. To simplify $n$ , find the largest $square factor$ of $n$ and pull its root outside.

ab = a b

Roots distribute over multiplication…

a + b \neq = a + b

…but NOT over addition. This is the number-one surd error.

1Find the largest square factor of 50: it is 25 (since

50 = 25 \times 2

50 = 252 = 52

Answer

52

Tip — You can only add or subtract $like$ surds (same number under the root), exactly as you can only collect like terms in algebra. $2 + 3$ will not simplify.

Multiplying surds and the key identity

Multiplication is straightforward: multiply the outside numbers together and the insides together. The single most useful fact is that a surd times itself removes the root: $a \times a = a$ .

a \times a = a

The move that clears surds from a denominator.

1Expand as normal (FOIL):

6 - 35 + 25 - 55

55 = 5

, so this is

6 - 35 + 25 - 5

3Collect:

(6 - 5) + (- 35 + 25) = 1 - 5

Answer

1 - 5

Rationalising the denominator

Convention says a surd should not be left in the denominator of a fraction. $Rationalising$ removes it. For a single surd on the bottom, multiply top and bottom by that surd — because $a \times a = a$ turns the denominator rational. Multiplying by $\frac{a}{a}$ (which is just 1) never changes the value.

When the denominator is $a \pm b$ , multiply by its $conjugate$ , $a \mp b$ . The difference of two squares, $(a + b) (a - b) = a^{2} - b$ , wipes out the surd in one stroke.

\frac{1}{a} = \frac{1}{a} \times \frac{a}{a} = \frac{a}{a}

Single surd: multiply by that surd over itself.

\frac{1}{a + b} \times \frac{a - b}{a - b} = \frac{a - b}{a ^{2} - b}

Two terms: multiply by the conjugate.

1Multiply top and bottom by

3

\frac{6}{3} \times \frac{3}{3} = \frac{6 3}{3}

2Simplify:

\frac{6}{3} 3 = 23

Answer

23

Why this works

The conjugate works because of the difference of two squares $(a + b) (a - b) = a^{2} - b^{2}$ . Squaring a surd is exactly what destroys it, so pairing $a + b$ with $a - b$ is engineered to make the surds cancel.

Think like an examiner

“Give your answer in the form $a + b c$ ” means a fully simplified, rationalised surd — no roots in any denominator and $c$ as small as possible.
Marks are commonly split: one for choosing the correct conjugate, one for the difference-of-squares denominator, one for full simplification.
Always simplify surds ( $50 \to 52$ ) before combining — “unlike” surds often become “like” once simplified.
Keep answers exact throughout; a decimal such as 2.83 will lose the “exact form” marks even if numerically close.

Common misconceptions

Writing

a + b = a + b

(e.g.

9 + 16 = 3 + 4

Roots do not split over addition:

9 + 16 = 25 = 5

, not 7.

Trying to add unlike surds:

2 + 3 = 5

Only like surds combine.

2 + 3

stays as it is.

Rationalising

\frac{1}{a + b}

by multiplying by

\frac{b}{b}

Use the conjugate

a - b

, so the difference of two squares clears the surd.

Leaving

a \times a

a^{2}

without simplifying.

a \times a = a

— this is the whole point of rationalising.

Surd toolkit

ab = a b

Split a root over a product.

a a = a

A surd times itself is rational.

\frac{1}{a} = \frac{a}{a}

Rationalise a single surd.

(a + b) (a - b) = a^{2} - b

Conjugate: difference of squares.

Stretch yourself

A right-angled triangle has legs $8$ and $18$ . Find the exact length of the hypotenuse, fully simplified.

Hint — Use Pythagoras, but simplify $8$ and $18$ first — or note that squaring a surd removes the root entirely.

Questions students ask

What exactly is a surd?

A surd is a root of a whole number that is irrational — it cannot be written exactly as a fraction or terminating decimal, like

2

5

. Roots that simplify to whole numbers, such as

16 = 4

, are not surds.

Why do we rationalise the denominator?

Partly convention, partly practicality: a rational denominator makes fractions easier to add, compare and evaluate, and it is the standard “tidy” form examiners expect. Multiplying by the surd (or conjugate) over itself changes the appearance without changing the value.

How do I simplify a surd like

72

Find the largest square factor.

72 = 36 \times 2

, so

72 = 362 = 62

. If you miss the largest factor, simplify again until no square factors remain.

When do I use the conjugate?

Whenever the denominator has two terms with a surd, such as

3 - 2

. Multiply top and bottom by the conjugate

3 + 2

; the difference of two squares removes the surd from the bottom.

Key takeaways

A surd is an irrational root; keep it exact rather than rounding.
Simplify by extracting the largest square factor: $50 = 52$ .
Roots split over $\times$ but never over $+$ ; only like surds add.
Single-surd denominator: multiply by that surd over itself.
Two-term denominator $a \pm b$ : multiply by the conjugate (difference of two squares).

How this fits the course

Build on

Completing the Square

Leads to

Quadratic Functions

Test yourself

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