A surd is an irrational number expressed as the root of a whole number that cannot be simplified to remove the root sign. For example, the square root of 2 (approximately 1.4142...) is a surd because its decimal expansion never terminates or repeats.
To simplify a surd, find the largest perfect square factor of the number under the root sign. For instance, the square root of 72 can be simplified as the square root of (36 x 2), which equals 6 times the square root of 2. This process makes calculations cleaner and is a key skill tested in GCSE and A-level mathematics.
Multiplication: The square root of a multiplied by the square root of b equals the square root of (a x b). Division: The square root of a divided by the square root of b equals the square root of (a / b). Rationalising the denominator: To remove a surd from the denominator, multiply both numerator and denominator by the surd.
Surds provide exact values rather than rounded decimal approximations. In engineering, architecture, and physics, maintaining exact values through calculations prevents rounding errors from compounding. GCSE and A-level exam boards expect students to express answers in surd form where appropriate.
Data verified against official UK government sources. Last checked April 2026.