Simplifying surds
Objective. Extract square factors to simplify irrational roots.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Simplifying a surd means rewriting a root so the number under the root sign has no remaining square factors. The method is to split the number into a square factor and another factor, then take the square root of the square factor outside the root sign, since √(ab) = √a × √b. For example, √50 = √(25 × 2) = √25 × √2 = 5√2. Always look for the largest square factor first to simplify in one step rather than several.
Worked example
Simplify the surd √72 fully, giving your answer in the form a√b.
- 1Find the largest square factor of 72: 36 × 2 = 72.
- 2Write √72 = √36 × √2.
- 3Evaluate √36 = 6.
- 4Combine to give 6√2.
Answer: 6√2
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Practice
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Start practisingCommon mistake. Pupils sometimes split a number using a non-square factor, such as √50 = √10 × √5, which cannot be simplified further. Always check the chosen factor pair includes a perfect square like 4, 9, 16 or 25.
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