Ahh, indices and surds, they look so messy and many find them hard to deal with… but they’re perhaps some of the easiest marks in your A Level paper.

### Indices

Let’s begin with the basics: An index (indices is the plural) is the power to which something is raised, for example in the expression 2^{3} the index is 3 and 2 is referred to as the base.

#### Rules of indices

Rule | Example |
---|---|

### Rational and irrational numbers

A rational number is one which can be written as either can integer or as the ratio of two integers (i.e. a fraction). For example, 2, , .

An irrational number is one which *cannot be expressed* as a fraction, for example or these numbers do not end or have a recurring pattern.

### Surds

Surds are expressions for irrational numbers, for example

#### Simplifying surds

To simplify a surd, such as firstly write it as the product of other integers, in this case . This can be expressed as . You know the square root of 9 is 3 and hence can simplify the expression further:

##### Examples

Simplify the following surds:

**1.**

**2.**

**3.**

**4.**

Here are the following worked solutions:

**1.** = = =

**2.** = = =

**3.** = = = =

**4.** =

#### Rationalising the denominator

Knowing (x + y)(x – y) = x^{2} – y^{2} can be used to solve more complex surds

For example:

Simplify the following surd:

By multiplying both sides by you can rationalise the denominator (i.e. turn it into a rational number):

Using the rule mentioned above:

You can then expand the numerator to give your final answer: