MathematicsExam code: 0580
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
Worked example
Multiplying and dividing surds with numbers in front
Work out 2√5 × 3√20 and 12√18 ÷ 4√2. Both answers are whole numbers.
Solution:
- Deal with the whole numbers and the roots separately
- 2 × 3 = 6 and √5 × √20 = √100 = 10
- So 2√5 × 3√20 = 6 × 10 = 60
- For the division, 12 ÷ 4 = 3 and √18 ÷ √2 = √9 = 3
- So 12√18 ÷ 4√2 = 3 × 3 = 9
- Simplifying a surd first gives the same answer: √20 = 2√5, so 2√5 × 6√5 = 12 × 5 = 60
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