MathematicsExam code: 0580

Surds

Number

PractiseFlashcards

Taking out a square factor

  • Look for a factor of the number that is a perfect square, then split the root across the factor pair
  • Choosing the largest square factor finishes the job in one step
    • 48 = 16 × 3, so √48 = √16 × √3 = 4√3
Two simplifications driven by spotting a square factor: root 8 becomes root 4 times root 2, which is 2 root 2, and root 720 becomes root 144 times root 5, which is 12 root 5
Source: Surds & exact values by Save My Exams
  • A smaller square factor still works but leaves more to do afterwards

Collecting after simplifying

  • Where several surds appear, simplify each one first: terms that looked unlike often become like
    • √75 = 5√3 and √12 = 2√3, so √75 + √12 = 7√3

Expanding brackets

  • Brackets containing surds expand exactly as algebraic brackets do, then simplify using (√a)² = a
    • (2 + √5)(3 − √5) = 6 − 2√5 + 3√5 − 5, which collects to 1 + √5

Worked example

Expanding brackets that contain surds

Expand and simplify (4 + √3)(5 − 2√3), giving your answer in the form a + b√3.

Solution:

  • Multiply each term in the first bracket by each term in the second, keeping every sign
  • 4 × 5 = 20 and 4 × (−2√3) = −8√3
  • √3 × 5 = 5√3 and √3 × (−2√3) = −2 × (√3)² = −2 × 3 = −6
  • Write all four terms down: 20 − 8√3 + 5√3 − 6
  • Collect the whole numbers and the root terms separately: 20 − 6 = 14 and −8√3 + 5√3 = −3√3
  • The answer is 14 − 3√3, so a = 14 and b = −3

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