Surds and quadratic functions

A surd is an irrational number written using a root sign, such as √2, √7 or ∛5. Irrational means the number cannot be written as a fraction p/q of two integers, so its decimal expansion never ends and never repeats. Not every root is a surd: √49 = 7, √0.25 = 0.5 and ∛8 = 2 are all rational, so they are not surds.

A quick test for square roots is to ask whether the number under the sign (the radicand) is a perfect square. If it is a perfect square of a rational number, the root is rational; if not, it is a surd.

Surds matter in Mathematical Methods because they let you write exact answers. A calculator gives √2 ≈ 1.414213…, but any decimal is a rounded approximation. When a question says 'give an exact answer', you must leave the surd in its simplest form rather than converting it to a decimal.

The key rules for square roots, valid for a, b ≥ 0, are:

• √a × √b = √(ab)
• √a ÷ √b = √(a/b), with b > 0
• (√a)² = a

Note that √(a + b) is not equal to √a + √b. For example √(9 + 16) = √25 = 5, but √9 + √16 = 7.

To simplify a surd, write the radicand as a product of the largest perfect square factor and a remaining factor, then take the square root of the perfect square. A surd is in simplest form when the radicand has no perfect square factor other than 1. It helps to know the perfect squares 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 by heart.

If you choose a perfect square factor that is not the largest, you will need to simplify again, so always check your final radicand.

Comparing surds this way avoids a calculator: square each value (or write it as an entire surd) and compare the radicands. The same idea lets you decide quickly whether an estimated answer is reasonable, because √72 must lie between √64 = 8 and √81 = 9.