A surd keeps an irrational root exact. Take out perfect-square factors: for nonnegative a,b, sqrt(ab)= a b. Simplify each root before collecting like terms. Example sqrt(72)=sqrt(36 x 2)=6 2, so sqrt(72)+sqrt(18)=6 2+3 2=9 2. 2 3+5 2 has unlike roots and does not combine. Also sqrt(9+16)=5, while 9+sqrt(16)=7: a root does not distribute over addition. Keep surds through working when an exact answer is required. Converting to decimals too early loses exactness.

A surd keeps an irrational root exact. Take out perfect-square factors: for nonnegative \(a,b\), \(\sqrt{ab}=\sqrt a\sqrt b\). Simplify each root before collecting like terms.

\(\sqrt{72}=\sqrt{36\times2}=6\sqrt2\), so \(\sqrt{72}+\sqrt{18}=6\sqrt2+3\sqrt2=9\sqrt2\).

\(2\sqrt3+5\sqrt2\) has unlike roots and does not combine. Also \(\sqrt{9+16}=5\), while \(\sqrt9+\sqrt{16}=7\): a root does not distribute over addition.

Keep surds through working when an exact answer is required. Converting to decimals too early loses exactness.

Surds

Cambridge IGCSE Mathematics 0580. Initial catalogue covers the assessed demands of February/March 2026 Paper 22; further Core and Extended mappings require paper-level scope review.

Exact roots and like surds

Exact roots and like surds