GCSE Maths: Surds and Bounds Guide

What is a surd?

A surd is a root (like √2) that cannot be simplified to a whole number, and is left in root form to keep the answer exact rather than rounding it.

Simplifying surds

To simplify a surd, look for a square factor: √12 = √(4 × 3) = 2√3. To multiply surds, multiply the numbers under the root: √2 × √3 = √6.

Rationalising the denominator

To remove a surd from the denominator of a fraction, multiply top and bottom by that surd, e.g. 1/√2 becomes √2/2 after multiplying by √2/√2.

Upper and lower bounds

A rounded measurement has an upper and lower bound — for example, a length given as 12 cm to the nearest cm could actually be anywhere from 11.5 cm up to (but not including) 12.5 cm.

Common mistakes

• Trying to simplify a surd without checking for square factors.
• Forgetting to rationalise a denominator when asked to give an answer in surd form.
• Using the rounded value instead of the correct bound when calculating a maximum or minimum possible answer.