B: Algebra and functions: Use and manipulate surds, including rationalising the denominator.

The idea

A surd is an irrational root such as √2 or √5. Simplify by extracting the largest square factor: √ab = √a × √b, so √50 = √(25×2) = 5√2. Combine like surds: 3√2 + 5√2 = 8√2. Multiply using √a × √b = √(ab), and expand brackets normally, e.g. (2+√3)(2−√3) = 4 − 3 = 1. Rationalising the denominator rewrites a fraction so the denominator is rational. For a single surd, multiply numerator and denominator by that surd: 1/√2 = √2/2. For a denominator a ± √b, multiply numerator and denominator by the conjugate a ∓ √b, using the difference of two squares. This skill gives exact answers in geometry, trigonometry and calculus.

What earns the marks

• Simplify surds by extracting the largest square factor, e.g. √72 = 6√2, and combine like surds through addition and subtraction.
• Multiply and divide surds, including expanding brackets such as (3+√2)(3−√2) = 7, and simplify the result.
• Rationalise a denominator containing a single surd by multiplying numerator and denominator by that surd, e.g. 5/√3 = 5√3/3.
• Rationalise a denominator of the form a ± √b by multiplying numerator and denominator by the conjugate, and simplify the resulting expression.
• Use surds in contextual problems, such as finding exact lengths in right-angled triangles or simplifying expressions in coordinate geometry.

Common mistakes and how to put them right

• Writing √(a+b) = √a + √b, e.g. √(9+16) = 3+4 = 7 instead of 5.

  Put it right: The square root of a sum is not the sum of the square roots; evaluate the sum first or leave as a single surd.

• Multiplying only the numerator by the conjugate when rationalising, e.g. writing 1/(2+√3) = 2−√3 without multiplying the denominator.

  Put it right: Multiply both numerator and denominator by the conjugate: 1/(2+√3) = (2−√3)/((2+√3)(2−√3)) = (2−√3)/1 = 2−√3. The result is correct here, but the method must apply to both parts to be valid in general.

• Simplifying √(a² + b²) as a + b, e.g. √(3²+4²) = 3+4 = 7 instead of 5.

  Put it right: This is a common error in Pythagoras; evaluate the sum of squares first, then take the square root.

• Incorrectly expanding brackets with surds, e.g. (√2+√3)² = 2+3 = 5 instead of 5+2√6.

  Put it right: Use (a+b)² = a² + 2ab + b², remembering that √2×√3 = √6.

Exam tips

• Always simplify surds fully before combining; look for the largest square factor to avoid multiple steps.
• When rationalising a denominator with two terms, multiply by the conjugate and show the expansion clearly, as method marks are often awarded for the correct conjugate.
• Check whether the question requires an exact answer in surd form; do not convert to a decimal unless asked, as this may lose accuracy marks.
• For expressions like (a+√b)(a−√b), recognise the difference of two squares to simplify quickly to a² − b.