Surds

In a nutshell

A surd is the square root of a number that results in an irrational number. Writing a number in surd form keeps the values accurate for a calculation, as the irrational numbers then do not need to be rounded to the nearest decimal. There are different rules for manipulating expressions with surds, including how to simplify, add and subtract them.

Definition

A surd is a root that cannot be simplified into a whole or rational number. They are irrational numbers, which cannot be written as a fraction and their decimal form goes on forever.

Example 1

Is $$\sqrt2$$ a surd?

$$\sqrt2$$ is an irrational number.​

$$\sqrt2=1.41421356237...$$ 

Therefore, yes, $$\sqrt2$$ is a surd.

Example 2

Is $$\sqrt9$$ a surd?

$$\sqrt9 = 3$$ is an integer.

Therefore, no, $$\sqrt{9}$$ is not a surd.

​

The rules of surds

These are the rules for surds which are derived from the laws of indices:

$$\boxed{\begin{aligned}\sqrt{a}\times \sqrt{b} = \sqrt{ab}\\\\\sqrt{a}\div \sqrt{b} = \sqrt{\frac{a}b}\\\\\sqrt{a}\times \sqrt{a} = a\end{aligned}}$$​​

Example 3

Simplify $$\sqrt{2}\times \sqrt8$$.

​$$\begin{aligned}\sqrt{2}\times \sqrt8&= \sqrt{16} \\&=\underline{4}\end{aligned}$$​​

Example 4

Simplify $$\sqrt{200} \div \sqrt8$$.

$$\begin{aligned}\sqrt{200} \div \sqrt8&= \sqrt{25} \\&=\underline{5}\end{aligned}$$​​

Simplifying surds

A surd is in its simplest form when the number within the root has no square numbers as factors. So when simplifying a surd, you are extracting the square numbers from underneath the root, leaving the smallest possible number.

Procedure

Example 5

Simplify $$\sqrt{20}$$.

$$4$$ is a square number and a factor of $$20$$.

$$\begin{aligned}\sqrt{20}&= \sqrt{4\times 5} \\&=\underline{2\sqrt5}\end{aligned}$$​​

Example 6

Simplify $$\sqrt{300}+\sqrt{27}$$.

$$100$$ is a square factor of $$300$$ and $$9$$ is a square factor of $$3$$.​

$$\begin{aligned}\sqrt{300}+\sqrt{27}&= \sqrt{100\times3}+\sqrt{9\times3} \\&=10\sqrt3+3\sqrt3\\&=\underline{13\sqrt3}\end{aligned}$$