What is a surd?
A surd is an irrational root that cannot be simplified to a whole number, such as √2 or √3. We keep surds in exact form rather than rounding. The surd laws let us combine and simplify them.
Key formula
√a × √b = √(ab)
√a ÷ √b = √(a/b)
(√a)2 = a
Rationalising: 1/√a = √a / a
Simplifying surds
Split the number under the root into a perfect-square factor times the rest, then take the square root of the perfect square outside:
- √8 = √(4 × 2) = 2√2
- √50 = √(25 × 2) = 5√2
- √27 = √(9 × 3) = 3√3
Only like surds (the same number under the root) can be added or subtracted: 2√3 + 5√3 = 7√3.
Worked example
Simplify √27 + √12.
√27 = 3√3 and √12 = √(4 × 3) = 2√3. Since both are like surds, add the numbers in front: 3√3 + 2√3 = 5√3.
Also √3 × √12 = √36 = 6, showing that a product of surds can be a whole number.
Rationalising the denominator
We usually do not leave a surd in a denominator. Multiply top and bottom by that surd: 6/√3 = (6 × √3)/(√3 × √3) = 6√3/3 = 2√3.
Remember
- Only like surds can be added or subtracted.
- √a × √a = a, a whole number.
- Always simplify surds fully before comparing them.