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Surds

Number

Multiplying and dividing

  • Roots multiply and divide by combining what sits underneath
    • √a × √b = √(ab), so √6 × √10 = √60
    • √a ÷ √b = √(a/b), so √98 ÷ √2 = √49 = 7
  • The reverse is just as useful: a root can be split across a factor pair, which is what makes simplifying possible

Adding and subtracting

  • Only like surds combine, in the same way that only like terms collect in algebra
    • 5√3 + 2√3 = 7√3
  • Unlike surds stay as they are: √2 + √5 cannot be shortened
  • Numbers under different root signs never combine by addition
    • √9 + √16 is 3 + 4 = 7, which is not √25

Squaring a surd

  • Squaring undoes the root, so (√a)² = a
  • This is the property that clears roots from a denominator later
Worked example

Simplifying and then collecting

Simplify √50 + 3√8, giving your answer in the form a√b.

Solution:

  • Simplify each surd separately by taking out the largest square factor
  • 50 = 25 × 2, and 25 is a square number, so √50 = √25 × √2 = 5√2
  • 8 = 4 × 2, so √8 = √4 × √2 = 2√2
  • That makes 3√8 = 3 × 2√2 = 6√2
  • Both terms now have the same root, so they can be collected like terms
  • 5√2 + 6√2 = 11√2
  • Adding the numbers under the roots would give √58, which is wrong: √50 + √8 is not √58