0580

Surds

Number

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

Simplifying and collecting

Write √75 + √12 in the form a√b, where a and b are integers and b is as small as possible.

Solution:

  • Take the largest square factor from each: 75 = 25 × 3 and 12 = 4 × 3
  • √75 = √25 × √3 = 5√3
  • √12 = √4 × √3 = 2√3
  • Both are now multiples of √3, so they collect
  • 5√3 + 2√3 = 7√3