IGCSE Maths / Advanced algebra
Simplifying surds revision notes
Edexcel International GCSE Maths (4MA1), chapter 23: Advanced algebra, specification point 1.4a. Free lesson.
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A surd is a root that cannot be written exactly as a whole number or fraction. Simplifying surds and matching their roots lets you give exact answers and combine like terms.
What makes a root a surd
A surd is a root that cannot be written exactly as a whole number or a fraction. For example, √9 = 3, so √9 is not a surd, but √2 is. Keeping a root such as √2 is an exact answer. Writing it as 1.414 is only a rounded value, so it is not exact. √2 is a surd. √9 is not a surd because it is exactly 3.
How to simplify a surd fully
Write a simplified surd with a whole number outside the root and the smallest possible number inside it. To do this, take out the largest square factor that divides into the number under the root. This works because the root of a product can be split into the product of its roots. For example, √72 = √(36 × 2) = √36 × √2 = 6√2. Check the number left under the root. If any square number still divides it, simplify again.
How to add or subtract surds
Surds combine like terms in algebra: you can add or subtract them only when their roots match. First simplify each surd, because different-looking roots may become the same. If the roots still do not match, leave the terms separate. √12 + √27 = 2√3 + 3√3 = 5√3. Once the roots match, add or subtract the numbers in front and keep the root unchanged.
Why roots do not split across addition
A root does not split across a plus or minus sign. For example, √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. Since 7 ≠ 5, √a + √b is not the same as √(a + b). Roots split across multiplication and division, not addition or subtraction. Simplify surds before adding or subtracting them. Do not combine terms if their roots do not match.