Surds and rationalising denominators

What is a surd?

A surd is a root that cannot be simplified to a whole number or a simple fraction. For example, √2, √3, √5, √7 are surds; √4 = 2 is not (because it gives a rational answer).

Surds give exact answers. In CCEA examinations, you will often be asked to "leave your answer as a surd" or "in exact form" — this means do not use a calculator to convert to a decimal.

Simplifying surds

The product rule: √(ab) = √a × √b.

To simplify a surd, find the largest perfect-square factor:

• √72 = √(36 × 2) = √36 × √2 = 6√2.
• √50 = √(25 × 2) = 5√2.
• √48 = √(16 × 3) = 4√3.
• √200 = √(100 × 2) = 10√2.

The quotient rule: √(a/b) = √a / √b.

Adding and subtracting surds

You can only add or subtract like surds (same number under the root), just like collecting like terms.

• 3√2 + 5√2 = 8√2.
• 7√3 − 2√3 = 5√3.
• √8 + √18 = 2√2 + 3√2 = 5√2 (simplify each first!).

Multiplying surds

Use the product rule and expand brackets:

• √3 × √3 = 3 (a surd times itself gives the number under the root).
• √2 × √8 = √16 = 4.
• (3 + √5)(3 − √5) = 9 − 5 = 4 (difference of two squares pattern).
• (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3.

Rationalising the denominator

A fraction with a surd in the denominator is not in its simplest form. Rationalising means rewriting it with a rational (non-surd) denominator.

Type 1 — monomial denominator: multiply numerator and denominator by the surd. $$\frac{5}{\sqrt{3}} = \frac{5 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{5\sqrt{3}}{3}$$

Type 2 — binomial denominator (Higher): multiply by the conjugate (change the sign of the surd term). $$\frac{4}{3 + \sqrt{2}} = \frac{4(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})} = \frac{4(3 - \sqrt{2})}{9 - 2} = \frac{4(3 - \sqrt{2})}{7}$$

CCEA examiner context

CCEA Paper 1 (non-calculator) regularly tests surds since calculators cannot be used. You must know how to simplify surds, collect like surds, expand brackets with surds, and rationalise. Exact answers using surds also appear in Pythagoras and trigonometry questions.