## Surds

Surds are irrational numbers left in their root form. Most of the time this is in the form $\sqrt{x}$, although occasionally it can be $\sqrt[3]{x}$.

The reason for this is that it is much easier to write numbers in this form, as opposed to decimals that continue forever.

For example, to answer the equation, $x^2 = 2$.

Our answers could either be $\pm \sqrt{2}$ or $\pm 1.414213562$…, which do you think is simpler? You will encounter Surds a lot when solving Quadratic Equations.

Did you know? – The story goes that when one of Pythagoras’ students discovered that $\sqrt{2}$ was an irrational number, Pythagoras ordered for him to be drowned!

## Operations with Surds

### Multiplication of Surds

In general:

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

Basically, if you have two surds multiplied together, then you can combine the two by multiplying together the two numbers under the square root sign.

A few more examples:

$\sqrt{5} \times \sqrt{7} = \sqrt{35}$

As 35 is not a square number, this cannot be simplified any more.

$\sqrt{12} \times \sqrt{15} = \sqrt{12 \times 15} = \sqrt{180}$

You may notice that we can simplify the last answer even more.

$\sqrt{180} = \sqrt{4} \times \sqrt{5} \times \sqrt{9} = 2 \times \sqrt{5} \times 3 = 6\sqrt{5}$

It’s important to notice here that if the number under the square root sign contains a factor that is a square number, then we can simplify the surd by taking the square root of this factor outside of the square root sign.

A more complicated example.

$(1- \sqrt{2})(2-\sqrt{8})$

Although this may look complicated, we multiply this as we would any other set of brackets.

$(1- \sqrt{2})(2-\sqrt{8})$
=
$2-\sqrt{8} -2\sqrt{2} + \sqrt{2}\sqrt{8}$
=
$2-2\sqrt{2} -2\sqrt{2} + \sqrt{16}$ (Note that $\sqrt{8} = 2\sqrt{2}$)
=
$2-4\sqrt{2} + 4$
=
$6-4\sqrt{2}$

### Addition and Subtraction of Surds

When adding and subtracting surds we can treat them as if we were adding $x$s and $y$s. If the surds are the same then we can group them, if not then there is nothing we can do.

For example – $23\sqrt{5} - 13\sqrt{5} = 10\sqrt{5}$

However, $23\sqrt{5} - 13\sqrt{7} = 23\sqrt{5} - 13\sqrt{7}$ cannot be simplified because the number under the square roots are different, ($\sqrt{5}$ and $\sqrt{7}$).

### Rationalising the Denominator

Mathematicians like things to be neat, and fractions containing a surd denominator are considered untidy. This little problem can be solved by multiplying the top and the bottom of the fraction by a particular expression, removing the surd from the bottom of a fraction.

Fractions in the form $\frac{a}{\sqrt{b}}$, multiply both top and bottom by $\sqrt{b}$

Example – Rationalise the fraction $\frac{1}{\sqrt{5}}$

$\frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5}$

A more complicated example

Fractions in the form $\frac{a}{b + \sqrt{c}}$, multiply top and bottom by $b - \sqrt{c}$

Example – Rationalise the fraction $\frac{3}{4 + \sqrt{3}}$

In these example you change the sign in front of the surd in the denominator of the fraction, doing so removes the square root from the bottom of your fraction.

$\frac{3(4 - \sqrt{3})}{(4 + \sqrt{3})(4 - \sqrt{3})}$
=
$\frac{3(4 - \sqrt{3})}{16 +4\sqrt{3}- 4\sqrt{3} -3}$
=
$\frac{3(4 - \sqrt{3})}{13}$ or $\frac{12 - 3\sqrt{3}}{13}$