Surds · Year 12
In Year 12 you learn: rationalising the denominator.
A surd is a square root that does not come out as a whole number, like √2 or √12. Leaving answers as surds keeps them exact instead of rounding. We simplify surds by taking out square factors, and rationalise denominators so that no root is left on the bottom of a fraction.
Key facts
- √(ab) = √a × √b and √(a/b) = (√a)/(√b).
- √a × √a = a.
- Simplify using the biggest square factor: √72 = √36 × √2 = 6√2.
- Only like surds can be added: 3√5 + 2√5 = 5√5, but √2 + √3 stays as it is. √a + √b ≠ √(a + b).
- Rationalise k/(√a) by multiplying top and bottom by √a.
- Rationalise k/(a + √b) by multiplying top and bottom by (a − √b) (difference of two squares).
Worked examples
Example 1
Rationalise the denominator of 10/(√5).
Answer: 2√5
Example 2
Rationalise and simplify 6/(3 − √3).
Answer: 3 + √3