Draw a square with sides of 1 cm and measure the diagonal — it's √2 cm, a number whose decimal (1.41421356…) never stops and never repeats. A surd is a square root like this that doesn't come out as a whole number, such as √2 or √3. Because the decimal runs on forever, we leave it written as √2 — that surd form is the exact value.
Combining surds
Surds multiply and divide very freely — you can slide numbers in and out of the root sign — but adding is fussier.
- √a × √b = √(ab) — e.g. √3 × √12 = √36 = 6.
- √a ÷ √b = √(a/b).
- You can only add or subtract like surds: 2√3 + 4√3 = 6√3 (just like 2 apples + 4 apples = 6 apples). But √2 + √3 can't be combined — they're different "fruit".
| Rule | Works? | Example |
|---|---|---|
| √a × √b = √(ab) | always | √5 × √20 = √100 = 10 |
| √a ÷ √b = √(a/b) | always | √18 ÷ √2 = √9 = 3 |
| add/subtract | like surds only | 2√3 + 4√3 = 6√3 |
| √(a + b) = √a + √b | never | see warning below |
Simplifying a surd
To simplify, look for a square number (4, 9, 16, 25…) that divides into the number under the root, split the root, and take the square root outside. Pulling out the biggest square factor finishes it in one go.
Rationalising the denominator
By convention it's untidy to leave a surd on the bottom of a fraction. The trick is that √a × √a = a, so multiplying top and bottom by the offending surd clears it from the denominator without changing the value (you've really multiplied by 1).
Memory hook: surds are exact roots — multiply and divide freely, only add or subtract matching surds, simplify using square factors, and rationalise to clear surds from the bottom.