Surds (radicals) and Simplification

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.