Summary — The Whole Chapter at a Glance
A surd is the square root of a positive integer that isn't a perfect square (e.g. √2, √13 — but NOT √16, since that's just 4).
Surds let us write exact values instead of rounded decimals — this matters a lot for accuracy in exams.
You can multiply and divide numbers under square roots together: √a × √b = √(ab), and √a ÷ √b = √(a÷b).
You cannot add or subtract numbers under square roots — √9 + √4 ≠ √13. This is the single most common mistake in this topic.
Simplifying a surd means pulling out the largest perfect-square factor hiding inside it, e.g. √48 = 4√3.
You can add/subtract surds only once they're "like terms" — same number under the root — just like collecting algebraic terms.
Rationalising the denominator means rewriting a fraction so there's no surd left on the bottom.
For a single surd on the bottom, multiply top and bottom by that surd. For an expression like (a + √b), multiply by its "conjugate" (a − √b), using the difference of two squares.
1 · What Is a Surd, and Why Bother?
Picture your calculator. If you type in √16, it happily gives you 4 — clean, exact, done. But if you type in √2, it spits out something like 1.414213562... and keeps going forever. That's because √2 is irrational — it can never be written as a neat fraction or a terminating/repeating decimal. No matter how many digits your calculator shows you, it's always an approximation, never the exact value.
This is exactly the problem surds solve. Instead of writing the rounded, slightly-wrong decimal, we just leave the answer as √2. It looks less "finished" than a decimal, but it's actually more accurate — it's the exact value, with zero rounding error.
A surd = the square root of a positive integer that is NOT a perfect square
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... — their square roots are whole numbers, so they are NOT surds.
Spotting surds vs non-surds
This trips people up more than you'd expect, because it's not really about how "ugly" the number looks — it's about a precise definition.
| These ARE surds | Why |
|---|
| √2, √13, √99, √201 | None of 2, 13, 99, or 201 are perfect squares, so their square roots can't simplify to a whole number. |
| These are NOT surds | Why |
|---|
| √16 | 16 is a perfect square (4²), so √16 = 4 exactly — a whole number, not a surd. |
| √(2/3) | 2/3 isn't an integer at all — surds are specifically defined using integers. |
| √6.23 | 6.23 isn't an integer either — same reason. |
Think of it like this
A surd is basically a square root that "doesn't clean up." If a perfect square is hiding inside the number (even partially), we can rescue part of it — but if there's genuinely no perfect square factor at all, the whole thing stays trapped under the root sign forever, in its most simplified form.
Practice Question
Which of the following are surds: √25, √50, √(9/4), √0.5, √11?
▶ Show Answer
2 · The Four Legal Moves With Surds
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Move 1 — Multiplying surds