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MathidiaMathematics Academy
Surds, indices and exact algebra
Lesson 6 of 75
Introduction

Expanding surd brackets

Objective. Expand and simplify products involving two surds.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Expanding surd brackets means multiplying out expressions containing square roots using the same method as expanding algebraic brackets, then simplifying any resulting surds or combining like terms. Each term in one bracket is multiplied by each term in the other, using the rule √a×√b = √(ab), and any square of a surd, (√a)², simplifies to a. Like surd terms can then be collected, and any surd results should be simplified to their simplest form. For example, (√3+2)(√3−5) = 3 − 5√3 + 2√3 − 10 = −7 − 3√3.

Worked example

Expand and simplify (√5+3)(√5−2), giving your answer in the form a+b√5.

-1124679+3
Number line from -1 to 9 with a jump of 3 forwards from 5
  1. 1Multiply out: √5×√5 + √5×(−2) + 3×√5 + 3×(−2)
  2. 2Simplify each term: 5 − 2√5 + 3√5 − 6
  3. 3Combine the surd terms: −2√5 + 3√5 = √5
  4. 4Combine the integer terms: 5 − 6 = −1

Answer: −1+√5

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Common mistake. A common error is writing √a × √a as √(a²) and leaving it unsimplified, or forgetting to multiply every pair of terms; always multiply fully using FOIL and simplify (√a)² to a straightaway.

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