A surd is a number that involves a square root, cube root or higher root that cannot be simplified to a whole number. Common examples include \( \sqrt{2} \), \( \sqrt{5} \) and \( \sqrt{7} \). Surds are exact values, not approximations.

 

 

Simplifying surds

 

A surd can sometimes be simplified by factorising the number inside the root. Any factor that is a perfect square can be taken outside the root.

 

For example:

$$
\sqrt{12}
$$

 

Factorise \( 12 \):

$$
12 = 4 \times 3
$$

 

So:

$$
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\sqrt{3}
$$

$$
= 2\sqrt{3}
$$

 

 

Adding and subtracting surds

 

Surds can only be added or subtracted if they are like surds, meaning they have the same root part.

 

For example:

$$
3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}
$$

 

But:

$$
\sqrt{5} + \sqrt{3}
$$

cannot be simplified further because the surds are different.

 

Multiplying surds

 

When multiplying surds, multiply the numbers outside the roots and multiply the numbers inside the roots.

 

For example:

$$
\sqrt{2} \times \sqrt{8}
$$

 

Combine under one root:

$$
\sqrt{16}
$$

$$
= 4
$$

 

Another example:

$$
3\sqrt{2} \times 4\sqrt{5}
$$

$$
= 12\sqrt{10}
$$

 

 

Dividing surds

 

When dividing surds, divide the numbers outside the roots and divide the numbers inside the roots.

 

For example:

$$
\frac{\sqrt{18}}{\sqrt{2}}
$$

$$
= \sqrt{9}
$$

$$
= 3
$$

 

 

Surds and powers

 

Surds can also be written using indices. For example:

$$
\sqrt{5} = 5^{\frac{1}{2}}
$$

 

This can be useful when applying the rules of indices.

 

Manipulating and simplifying surds helps keep answers exact and is important when working with algebraic expressions, equations and further calculations involving roots.