How do you simplify surds?
Short answer
To simplify a surd, write the number under the root as a product that includes the largest possible square number, then take the square root of that square outside. For example, √50 = √(25 × 2) = 5√2. A surd is fully simplified when no square number other than 1 divides the number under the root.
Step by step
- List the square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Find the largest one that divides the number under the root.
- Split the root: √(a × b) = √a × √b.
- Take the square root of the square number and write it in front.
- Collect like surds if there is more than one term: 3√2 + 5√2 = 8√2.
Worked example
Simplify √72 + √18
- √72 = √(36 × 2) = 6√2
- √18 = √(9 × 2) = 3√2
- 6√2 + 3√2 = 9√2
Common mistakes
- Using a square number that is not the largest, for example √72 = 2√18, which is not fully simplified.
- Adding the numbers under the roots: √2 + √3 is not √5.
- Forgetting that √a × √a = a.
Useful for:GCSEWAECIGCSEMatric
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