Surds

La méthode, étape par étape

1. 1

   Find the largest square factor

   √72 → 72 = 36 × 2 → √36 × √2 = 6√2

   Splitting off the largest perfect square gets to the simplest form in one step. Using a smaller square factor works but takes more rounds: 72 as 4 × 18 gives 2√18, which still simplifies further.

2. 2

   Multiply surds by multiplying what is inside

   √3 × √12 = √36 = 6

   Roots multiply directly, and the result is often rational even when the parts are not. Checking whether the product under the root is a perfect square is worth doing before any other simplification.

3. 3

   Only add surds that are already alike

   3√5 + 4√5 = 7√5, but √2 + √3 stays as it is

   Like surds combine the way like terms in algebra do — the root behaves as the unit. Unlike surds cannot be added at all, and writing √2 + √3 = √5 is the most common error in the topic. Squaring both sides shows immediately that it is false.

4. 4

   Rationalise a simple denominator

   3/√2 = 3/√2 × √2/√2 = 3√2/2

   Multiplying top and bottom by the surd clears the root from the bottom without changing the value, because √2/√2 is 1. The convention predates calculators — dividing by an irrational number by hand was genuinely difficult — but it remains what exams expect.

5. 5

   Rationalise a two-term denominator with the conjugate

   1/(3+√2) × (3−√2)/(3−√2) = (3−√2)/7

   The conjugate changes the sign between the terms, and multiplying the two gives a difference of two squares — 9 − 2 = 7 — which is rational. This is the only reliable way to clear a denominator containing a sum.

Why leave it as a root at all

√2 written as a decimal is 1.41421356… and does not terminate or repeat, so any decimal version is an approximation. Leaving it as √2 keeps the value exact, and in a question with several steps that exactness prevents rounding errors from accumulating into a wrong final answer.

Not every root is a surd. √9 is 3, which is rational, so it is not a surd — the term applies only to roots that cannot be written as a fraction. Students sometimes treat every root symbol as a surd, which makes simplification look harder than it is.

• A surd is an irrational root left in exact form
• √a × √b = √(ab)
• Simplify by extracting the largest square factor
• Only like surds add: 3√5 + 4√5 = 7√5
• √2 + √3 does not equal √5

The error worth naming

Writing √a + √b = √(a+b) is the defining mistake of this topic, and it is tempting because the multiplication rule genuinely does work that way. The quickest refutation is arithmetic: √9 + √16 is 3 + 4 = 7, while √25 is 5. Two numbers that are not equal.

We show that example once and refer back to it whenever the error reappears. It is far more effective than restating the rule, because the student can check it themselves in five seconds.

Why rationalising is still required

The original reason was practical: before calculators, dividing by 1.414 was much harder than dividing by 2, so clearing the root from the denominator saved real effort. That reason has gone, and the convention has not.

It survives because it gives every expression a single canonical form, which makes answers directly comparable and marking unambiguous. Exam boards expect it, so it is worth doing regardless of whether the original motivation still applies.

How we teach surds

We drill perfect squares to 15² first. Simplifying surds is mostly the skill of spotting that 72 contains 36, and a student who has to search for square factors will miss the largest one and produce answers that are not fully simplified.

We teach the conjugate as difference of two squares rather than as a separate rule, because students have already met that identity in algebra. Connecting the two means one technique to remember instead of two.