# Surds

**Surds**

It is not possible to find the square roots of many numbers.

For example, √3 cannot be written exactly as a fraction or a decimal. This is because it is an **irrational number**.

Therefore, it is often better to write the number as a square root sign √3.

A **surd** is therefore – √3.

**Multiplying surds**

When multiplying surds, it is squaring the square root.

Therefore, √*a* × √*a* = *a*

Which can also be written as *a*√*a*

Also the following is also possible, √*a* × √*b* = √*ab*

Points to note:

- If “
*a”*and*b”*are irrational numbers their product can be a rational number

**Dividing surds**

When dividing surds, the following formula is used:

√*a* ÷ √*b* = √ *a/b*

Points to note:

- Similar to multiplication, if “
*a”*and*b”*are irrational numbers their product can be a rational number.

**Simplifying surds**

To simplify surds it is written in the form *a***√***b*.

For example, simplify **√**75 by writing it in the form *a***√***b*.

Start by finding the largest square number that divides into 75.

This is 25. We can use this to write:

**√**75 = **√**(25 × 3)

= **√**25 × **√**3

= 5**√**3

**Adding and subtracting surds**

Adding and subtracting surds is possible if the number under the square root sign is the same.

For example, simplify √27 + √75.

Start by writing √27 and √75 in their simplest forms.

√27 = √(9 x 3)

= √9 x √3

= 3√3

√75 = √(25 x 3)

= √25 x √3

= 5√3

√27 + √75 = 3√3 + 5√3 = 8√3

**Rationalizing the denominator**

**Rationalising the denominator** is when a fraction has a surd as the denominator and is written so that the denominator is a rational number.

For example:

Here, the numerator and denominator is multiplied by √2 to make the denominator into a whole number. The denominator is then normally re-writen as a rational number.