Surds - Simplifying and Manipulating Worksheet

An AQA Level 2 Further Maths (8365) worksheet on simplifying and manipulating surds, covering surd rules, combining like surds, and expanding brackets, with full worked answers.

A surd is a root (like √2 or √18) that can’t be simplified to a whole number — this worksheet builds the core skills for handling them algebraically. It opens with two worked examples: simplifying a surd by finding its largest square factor, and combining “like surds” by adding or subtracting coefficients once every term is fully simplified. From there it moves through three differentiated tiers: Fluency (simplifying surds fully), Reasoning (combining and multiplying surds, including questions where terms look unlike but simplify to the same root), and Problem Solving (expanding single and double brackets involving surds, plus two contextualised area/perimeter questions).

The final problem-solving questions introduce the difference of two squares with surds — showing students why expressions like (√5 + 2)(√5 − 2) always cancel to a rational number, a result they’ll rely on again when rationalising denominators. Full worked solutions are included on the final pages, so this works for independent study, homework, or self-marking in class.

This is the first in a two-part surds sequence (Surds I: simplifying and manipulating), aimed at high-attaining Year 10/11 students taking the AQA Level 2 Certificate in Further Mathematics (8365) alongside GCSE, and at students bridging into A-level Maths, where fluency with surds is assumed from the first week.

Two PDF versions are included: a standard printable copy with name/class/date fields, and a name-free version for digital display.

Last updated: 12th July 2026.