Exact before approximate.

A surd represents an irrational root exactly. Factor out square factors, combine like radicals and expand products with the distributive law. For example, ( $\sqrt{3}$ + 2)² = 7 + 4 $\sqrt{3}$ ; the middle term must be included.

Rationalising a denominator changes the form, not the value. Multiply numerator and denominator by the same non-zero expression. For a binomial denominator, a conjugate uses (a + b)(a − b) = $a^{2}$ − $b^{2}$ . Round only when a numerical approximation is requested.

INTERACTIVE MODEL

Try it. Watch it change.

Area / square units8

\sqrt{8}

= 2

\sqrt{2}

≈ 2.8284

A square’s side length is the positive square root of its area. Exact surd form avoids rounding.

Explore: Find two square areas whose side lengths are different multiples of $\sqrt{2}$ .

Difference of squares

(a+b)(a-b)=a^2-b^2

WORKED EXAMPLE

Rationalise $\frac{1}{\sqrt5-1}$ .

Multiply by $\frac{\sqrt5+1}{\sqrt5+1}$ .
Denominator = 5 − 1 = 4.
The exact result is $\frac{\sqrt5+1}{4}$ .

Assumed knowledge

Index laws, square factors and expanding brackets.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Simplify surds and rationalise a denominator.
Retain exact values through a calculation.

QCAA Mathematical Methods 2025 v1.3 · p. 16 ↗

READY TO TRY IT?

Exact before approximate.

Rationalise 15−1\frac{1}{\sqrt5-1}5​−11​.

Put the idea to work.

Rationalise $\frac{1}{\sqrt5-1}$ .