A surd is simply a square root (or other root) that doesn't work out to a nice whole number — like √2, √3, or √5. We leave them written as roots instead of rounding to a decimal.
Why? Because √2 = 1.41421356… goes on forever. Writing √2 is exact; writing 1.414 is only an approximation.
Simplifying surds: look for the largest perfect square that goes into the number under the root.
√50 = √(25 × 2) = √25 × √2 = 5√2
√72 = √(36 × 2) = 6√2
You can add and subtract surds that have the same number under the root sign (just like collecting like terms in algebra):
3√5 + 2√5 = 5√5 — but you CANNOT simplify 3√5 + 2√3 (different numbers under the root).
Multiplying surds: √a × √b = √(ab)
For example: √3 × √12 = √36 = 6
Removing surds from the bottom of a fraction (rationalising the denominator) is covered in 'Rationalising the Denominator (Higher Only)'.