Surds: Simplify, add and subtract

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 $\sqrt3$  a surd?

$\sqrt3 =1.73205080757...$ is an irrational number.

Therefore, $\sqrt{3}$ is a surd.

Example 2

Is $\sqrt4$ a surd?

$\sqrt4 = 2$ is a whole number.

Therefore, $\sqrt{4}$ is not a surd.

The rules of surds

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

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

$\sqrt{a}\div \sqrt{b} = \sqrt{\frac{a}b}$

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

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

1	Find square numbers that are factors of the number within the root.
2	Rewrite the surd as the product of the square number(s) and the other number(s).
3	Root the square number(s) and write it outside of the root, leaving the other number(s) inside.

Example 3

Simplify $\sqrt50$ .

 $25$ is a square number and a factor of $50$ .

$\begin{aligned}\sqrt{50}&= \sqrt{25 \times 2} \\&=\underline{5\sqrt2}\end{aligned}$

Adding and subtracting surds

You can add and subtract surds when the number within the roots are the same. For example, $\sqrt3$ and $2\sqrt3$ can be added or subtracted, but $\sqrt3$ and $2\sqrt5$ cannot.

procedure

1.	Simplify each surd.
2.	Collect the surds by adding or subtracting them.

Example 4

Simplify $\sqrt3 + 2\sqrt3 + 2\sqrt{5}$ .

The surds cannot be simplified any further. Find the like terms and combine them.

$\sqrt3 + 2\sqrt3 + 2\sqrt5 = \underline{3\sqrt3+2\sqrt5}$ 