SQA National 5 Mathematics

## What a surd is A **surd** is a root that cannot be written exactly as a whole number or fraction, so it is left in root form to keep it exact. $\sqrt{2}$, $\sqrt{7}$ and $\sqrt{50}$ are surds. $\sqrt{49}$ is not, because it equals 7. A decimal such as $\sqrt{2} = 1.41421\ldots$ never ends or repeats, and any rounded version is only an approximation, which is why an exact answer is written with the root sign. Two rules for multiplying and dividing roots do most of the work, and both need $a$ and $b$ to be positive: $$ \sqrt{a}\times\sqrt{b} = \sqrt{ab}, \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$ Also $\sqrt{a}\times\sqrt{a} = a$, which is what makes rationalising possible. There is no matching rule for addition: $\sqrt{9}+\sqrt{16} = 7$, but $\sqrt{9+16} = \sqrt{25} = 5$, so $\sqrt{a}+\sqrt{b}$ is not $\sqrt{a+b}$. ## Simplifying a surd To simplify $\sqrt{n}$, look for the largest square number that divides $n$, split the root into the square root of that number times the root of the rest, and evaluate the square root. $$ \sqrt{72} = \sqrt{36\times 2} = \sqrt{36}\times\sqrt{2} = 6\sqrt{2} $$ If the largest square factor is not obvious, a smaller one can be taken and the process repeated. $\sqrt{72} = \sqrt{4\times 18} = 2\sqrt{18} = 2\sqrt{9\times 2} = 6\sqrt{2}$ gives the same result. A surd is in its simplest form when the number under the root has no square factor other than 1. Another example: $\sqrt{50} = \sqrt{25\times 2} = 5\sqrt{2}$, and $3\sqrt{20}…