Surds

1.18 Surds – Cheat Sheet

Definition of a Surd

• A surd is an irrational root that cannot be simplified into a rational number.
• Examples: √2, √3, ³√5.
• Surds are left in root form for exact answers.

Basic Surd Rules

Simplifying Surds

• Break down the number inside the root into factors where one is a perfect square.
• √20 = √(4 × 5) = √4 × √5 = 2√5
• √200 = √(100 × 2) = 10√2
• √72 = √(36 × 2) = 6√2

Adding and Subtracting Surds

• Only like surds (same root value) can be added or subtracted.
• 3√2 + 5√2 = 8√2
• 7√3 − 2√3 = 5√3
• √2 + √3 cannot be simplified further.

Multiplying Surds

• Multiply numbers inside the root, then simplify.
• √2 × √8 = √16 = 4
• (2√3) × (5√2) = 10√6

Dividing Surds

• Divide numbers inside the root, then simplify.
• √50 ÷ √2 = √25 = 5
• (6√10) ÷ (2√5) = 3√2

Rationalising the Denominator

Case 1: Denominator is a simple surd

• Multiply top and bottom by the surd in the denominator.
• 5 / √3 = (5 × √3) / (√3 × √3) = 5√3 / 3

Case 2: Denominator is a binomial with a surd

• Multiply top and bottom by the conjugate (change the sign between terms).
• 1 / (√3 + 1) = (1 × (√3 − 1)) / ((√3 + 1)(√3 − 1))
  = (√3 − 1) / (3 − 1) = (√3 − 1) / 2

Examples Table