For a denominator containing a rational term and a surd, use its conjugate: keep the same two terms and reverse the sign between them. Their product is a difference of squares, so the surd terms cancel. Example 1 3-1=(3+1)/((3-1)(3+1))=(3+1)/(3-1)=(3+1)/(2). For 23+ 5, multiply both top and bottom by 3- 5: the result is (2(3- 5))/(9-5)=(3- 5)/(2). The conjugate must multiply the entire numerator. Check the denominator’s sign when its first term is a surd; (2-3)(2+3)=2-9=-7.

For a denominator containing a rational term and a surd, use its conjugate: keep the same two terms and reverse the sign between them. Their product is a difference of squares, so the surd terms cancel.

\(\frac1{\sqrt3-1}=\frac{\sqrt3+1}{(\sqrt3-1)(\sqrt3+1)}=\frac{\sqrt3+1}{3-1}=\frac{\sqrt3+1}{2}\).

For \(\frac2{3+\sqrt5}\), multiply both top and bottom by \(3-\sqrt5\): the result is \(\frac{2(3-\sqrt5)}{9-5}=\frac{3-\sqrt5}{2}\).

The conjugate must multiply the entire numerator. Check the denominator’s sign when its first term is a surd; \((\sqrt2-3)(\sqrt2+3)=2-9=-7\).

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.

Rationalising a two-term denominator

Rationalising a two-term denominator