## What is a Surd?

When there's a term with a root which we can't simplify, it can be called a surd. For example, $$\sqrt{3}$$ is a surd, but $$\sqrt{4}$$ isn't, as $$\sqrt{4}=2$$ which no longer contains a root. Cube (and higher) roots such as $$\sqrt[3]{5}$$ are also called surds, but we won't be talking about them here.
Surds can also be called radicals.

## Simplifying Surds

The most likely thing you'll want to do with a surd is try and simplify it. In general, the rule you need to know how to apply is $\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$ Let's try and use this with an example: \begin{align} \sqrt{3\times4}&=\sqrt{3}\times\sqrt{4} \\ &=\sqrt{3}\times2 \\ &=2\sqrt{3} \end{align} Like brackets, you don't need to write the $$\times$$ sign here, it is implied. It's likely that you won't be given an actual product inside the root, and rather just one number. Above, it would have been $$\sqrt{12}$$. The key from that stage is to try and find a factor of that number which is a square number. Above, 4 is square so we could simplify the $$\sqrt{4}$$ to $$2$$. This is the main aim!

## Worked example, starting easy

\begin{align} \sqrt{63}&=\sqrt{9\times7} \\&=\sqrt{9}\times\sqrt{7} \\&=3\times\sqrt{7} \\&=3\sqrt{7} \end{align}

## Worked example; getting tricky

Here we'll look at a harder example, $$\sqrt{108}$$. This requires more workings out, but is the same process. $$108$$ is a much bigger number, so it's less obvious what its factors are. Let's start by noticing that it's even, so $$2$$ is a factor. \begin{align} \tfrac{108}{2} &= 54 \\ \Rightarrow 108 &= 2\times54 \end{align} Similarly with $$54$$, \begin{align} \tfrac{54}{2} &= 27 \\ \Rightarrow 54 &= 2\times27 \\ \Rightarrow 108 &= 2\times2\times27 \\ \Rightarrow 108 &= 4\times27 \end{align} We can do a similar process with $$27$$, or we might know already that $$27 = 9\times3$$, and $$9$$ is a square number so that's great! We're left with: \begin{align} 108 &= 4\times9\times3 \\ \Rightarrow \sqrt{108} &= \sqrt{4\times9\times3} \\ &= \sqrt{4}\times\sqrt{9}\times\sqrt{3} \\ &= 2\times3\times\sqrt{3} \\ &= 6\times\sqrt{3} \\ &= 6\sqrt{3} \\ \text{So: } \sqrt{108} &= 6\sqrt{3} \end{align}

This skill can be useful in lots of areas. One place you'll often need it though is when using the quadratic formula.