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:

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.

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 $3\sqrt{20} = 3\sqrt{4\times 5} = 3\times 2\sqrt{5} = 6\sqrt{5}$.

Adding and subtracting surds

Surds can be added or subtracted only when they have the same number under the root, like terms in algebra. Simplify each one first, because two surds that look different may turn out to be alike.

Here $\sqrt{12} = \sqrt{4\times 3} = 2\sqrt{3}$ and $\sqrt{27} = \sqrt{9\times 3} = 3\sqrt{3}$. An expression like $2\sqrt{5}+3\sqrt{2}$ cannot be combined any further.

Multiplying surds and brackets

Multiply the numbers outside the roots together, and the numbers inside the roots together, then simplify.

Brackets containing surds are expanded the same way as in algebra, with every term in the first bracket multiplying every term in the second.

A single term outside a bracket is handled in the same way: $\sqrt{3}(\sqrt{6}+\sqrt{3}) = \sqrt{18}+3 = 3\sqrt{2}+3$.

Rationalising the denominator

A fraction with a surd in the denominator is usually rewritten so that the denominator is a whole number. This is called rationalising the denominator. Multiplying the top and bottom by the same number leaves the value unchanged, and choosing that number to be the surd in the denominator removes the root from the bottom, because $\sqrt{a}\times\sqrt{a}=a$.

When the denominator is a number times a surd, multiply by the surd only. For $\dfrac{5}{2\sqrt{5}}$:

The answer is left as a fraction in its simplest form, with the surd on top. A fraction such as $\dfrac{\sqrt{7}}{\sqrt{2}}$ becomes $\dfrac{\sqrt{7}\times\sqrt{2}}{2} = \dfrac{\sqrt{14}}{2}$.

Exact values in context

Surds appear wherever a length comes from Pythagoras' theorem or a trigonometric ratio and the question asks for an exact answer. A right-angled triangle with shorter sides 3 cm and 4 cm has a hypotenuse of exactly 5 cm, but with shorter sides 2 cm and 4 cm the hypotenuse is $\sqrt{4+16} = \sqrt{20} = 2\sqrt{5}$ cm. The exact form $2\sqrt{5}$ is preferred to $4.47$ because nothing has been rounded. To compare sizes, the simplified form helps: $3\sqrt{2} = \sqrt{18}$, which lies between $\sqrt{16}=4$ and $\sqrt{25}=5$, so it is a little over 4.