CAMBRIDGE O LEVEL · 2025–2027

Surds.

Keep roots exact, simplify their structure and remove irrational denominators.

Syllabus 1.18Visual notesWorked examples

BY THE END

Simplify and combine surds · expand expressions · rationalise single- and two-term denominators

Surds keep irrational roots exact

Some roots are whole numbers, such as √9 = 3. Others cannot be written as exact fractions, such as √2. Keeping such an irrational root in exact form gives a surd instead of a rounded decimal.

A surd can sometimes be simplified by separating a square factor. Since 12 = 4 × 3, its square root is √4 × √3 = 2√3. Look for the largest square factor so the remaining root cannot be simplified again.

A surd is an irrational root expression, such as √2 or √5. Leave it exact when an exact answer is required. √9 = 3 is not a surd after simplification.

Look for a square factor

√20 = √(4 × 5) = 2√5

Use √(ab) = √a × √b for non-negative a and b. Choose the largest square factor to simplify efficiently.

Useful square factors

49162536496481100

Try these factors inside a square root.

Each small square has area 5 and side √5. Two small sides make one large side.

WORKED EXAMPLE

Simplify √72 and √200.

√72 = √(36 × 2) = 6√2.
√200 = √(100 × 2) = 10√2.

Pause and explainWhy does √12 simplify to 2√3 rather than 6?

The square root applies to factors, not to halving a number. The exact check is (2√3)² = 4 × 3 = 12.

Combine like surds; multiply carefully

Think of √3 as a particular quantity, just as x can stand for a quantity in algebra. Two lots of √3 plus five lots of √3 give seven lots of √3. Different root parts cannot simply be collected together.

Simplify every root first, because expressions that look different may become like terms. Multiplication works through factors, so root products can often simplify after multiplying. Addition does not split a square root: √(a + b) is generally not √a + √b.

WORKED EXAMPLE

Simplify √200 − √32.

10√2 − 4√2 = 6√2

After simplifying, the root parts match. Subtract the coefficients. You cannot combine 2√3 + √5 into a single like-surd term.

WORKED EXAMPLE

Multiply √6 × √15.

√90 = √(9 × 10) = 3√10

WORKED EXAMPLE

Expand (√3 + 2)(√3 − 1).

Multiply each term in the first bracket by each in the second.

3 − √3 + 2√3 − 2 = 1 + √3

Pause and explainCan 2√5 + 3√5 be collected?

Yes. They are five lots of the same root, giving 5√5. But 2√5 + 3√2 has different root parts.

Rationalise a single-surd denominator

A denominator containing a root can make equivalent exact forms difficult to compare. Rationalising rewrites the fraction with a rational denominator while keeping its value unchanged.

The permitted move is to multiply numerator and denominator by the same non-zero quantity. With a single square root below, multiplying by that root creates its square in the denominator. Because the whole fraction was multiplied by 1, the value stays the same.

Rationalising means rewriting an equivalent expression with a rational denominator. Multiply the numerator and denominator by the same non-zero expression.

WORKED EXAMPLE

Rationalise 10√5.

10√5 × √5√5 = 10√55 = 2√5

√5 × √5 = 5. The multiplier √5√5 equals 1, so the value is unchanged.

WORKED EXAMPLE

Simplify 32√3.

3√32 × 3 = √32

Pause and explainWhy multiply both parts of 1/√3 by √3?

The denominator becomes √3 × √3 = 3, so the equivalent fraction is √3 over 3. Multiplying only the denominator would change the value.

Two terms: use the conjugate

If a denominator contains a sum or difference, multiplying by one root alone will not remove every surd. Instead use the same two terms with the opposite sign between them, called the conjugate.

This uses the difference-of-squares identity: (a + b)(a − b) = a² − b². The middle terms cancel. Expand the numerator carefully as well, and check the sign of the rational denominator before simplifying the final result.

For a denominator a + √b, use a − √b; for a − √b, use a + √b. This sign-changing partner is its conjugate.

(a + √b)(a − √b)= a² − bNo surd remains

WORKED EXAMPLE

Rationalise 2√3 − 1.

2√3 − 1 × √3 + 1√3 + 1 = 2(√3 + 1)3 − 1 = √3 + 1

The middle terms cancel when expanding the denominator: (√3 − 1)(√3 + 1) = 3 − 1 = 2.

WORKED EXAMPLE

Rationalise 12 + √3.

2 − √3(2 + √3)(2 − √3) = 2 − √34 − 3 = 2 − √3

Pause and explainWhat is the conjugate of 2 + √3, and what product does it give?

It is 2 − √3. Their product is 4 − 3 = 1, so the root disappears from the denominator.

Your turn. Keep the answer exact.

Keep answers exact unless a decimal is requested. First search for square factors, then decide whether terms can be collected or a denominator needs rationalising.

After rationalising, check that the numerator was multiplied too. A quick decimal comparison can confirm equivalence, but your final exact answer should retain the simplified surd rather than its rounded calculator display.

Simplify √45, √48 and √125.

Reveal worked answer

√45 = 3√5; √48 = 4√3; √125 = 5√5. Extract the largest square factor each time.

Simplify 3√12 + √27 − √75.

Reveal worked answer

3√12 = 6√3, √27 = 3√3 and √75 = 5√3. Total = (6 + 3 − 5)√3 = 4√3.

Expand and simplify (√5 + 1)(√5 − 1).

Reveal worked answer

This is a difference of squares: 5 − 1 = 4.

Rationalise 6√3.

Reveal worked answer

Multiply by √3√3 to get 6√33 = 2√3.

Rationalise 1√5 − 2.

Reveal worked answer

Multiply by √5 + 2√5 + 2. The denominator becomes 5 − 4 = 1, so the answer is √5 + 2.