Operations with surds
Objective. Add, subtract and multiply compatible surd expressions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Surds can be added, subtracted or multiplied once they are expressed with matching roots, treating √n like a variable term. Like surds combine directly, so 3√2 + 5√2 = 8√2, just as 3x + 5x = 8x, but √2 + √3 cannot be simplified further. Multiplying surds uses √a × √b = √(ab), so 2√3 × 4√5 = 8√15, multiplying coefficients and the numbers under the roots separately. Always simplify any resulting surd fully after multiplying.
Worked example
Simplify 5√3 × 2√6 fully, combining the surds correctly.
- 1Multiply the coefficients: 5 × 2 = 10.
- 2Multiply the numbers under the roots: 3 × 6 = 18.
- 3This gives 10√18, which needs simplifying since 18 = 9 × 2.
- 4√18 = 3√2, so 10√18 = 30√2.
Answer: 30√2
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils often add unlike surds incorrectly, writing √2 + √8 = √10. Instead, simplify √8 = 2√2 first, so √2 + 2√2 = 3√2.
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