2 Sep 2026 · 6.30–7.30pm · No recap (first session)
A surd is a square root (or other root) that can’t be simplified to a whole number or a neat fraction — the decimal goes on forever without repeating.
Why it matters: in exact-answer questions, National 5 wants surds left in surd form rather than rounded as a decimal, because rounding loses accuracy.
\[\sqrt{a} \times \sqrt{b} = \sqrt{ab} \qquad \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Everything in this topic comes from these two rules. There’s no need to memorise anything else — just apply these correctly.
The idea: pull out any perfect-square factor hiding inside the surd.
Perfect squares to know by heart: 4, 9, 16, 25, 36, 49, 64, 81, 100
Simplify √50
Simplify √72
(If Imogen picks a smaller factor first, e.g. 4 × 18, that’s fine — she’ll just need a second simplifying step: √4×√18 = 2√18 = 2×√9×√2 = 2×3√2 = 6√2. Same answer, just show her the shortcut of finding the largest factor first.)
Only “like surds” (same number under the root) can be combined — treat √2 like a variable, similar to combining like terms with x.
Example: 3√2 + 5√2 = 8√2 Example: √8 + √18 = 2√2 + 3√2 = 5√2 (simplify first, then combine)
The rule: we never leave a surd on the bottom of a fraction. To remove it, multiply top and bottom by that same surd (this doesn’t change the value, since you’re multiplying by a form of 1).
Rationalise \(\dfrac{1}{\sqrt{5}}\)
\[\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}\]
Rationalise \(\dfrac{6}{\sqrt{3}}\)
\[\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\]
(Point out the final simplifying step — students often stop too early and leave 6√3⁄3 instead of cancelling to 2√3.)
| Time | Activity |
|---|---|
| 5 min | Explain what a surd is and why exact form matters, using √4 vs √2 |
| 10 min | Teach the two rules, work through examples 1–2 (simplifying) together |
| 5 min | Imogen tries 2–3 simplifying questions solo |
| 10 min | Teach rationalising, work through examples 3–4 together |
| 10 min | Imogen tries 2–3 rationalising questions solo |
| 10 min | Mixed practice + set homework |
Simplifying: 1. √8 → 2√2 2. √45 → 3√5 3. √98 → 7√2 4. √12 + √27 → 2√3 + 3√3 = 5√3
Rationalising: 5. \(\dfrac{3}{\sqrt{2}}\) → \(\dfrac{3\sqrt{2}}{2}\) 6. \(\dfrac{10}{\sqrt{5}}\) → \(2\sqrt{5}\) 7. \(\dfrac{4}{\sqrt{6}}\) → \(\dfrac{4\sqrt{6}}{6} = \dfrac{2\sqrt{6}}{3}\)