Maths revision: Indices and surds including ‘rationalising the denominator’

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 23 the index is 3 and 2 is referred to as the base.

Rules of indices

RuleExample
Dynamic image 0Dynamic image 1
Dynamic image 2Dynamic image 3
Dynamic image 4Dynamic image 5
Dynamic image 6Dynamic image 7
Dynamic image 8Dynamic image 9
Dynamic image 10Dynamic image 11
Dynamic image 12Dynamic image 13

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, Dynamic image 14, Dynamic image 15.

An irrational number is one which cannot be expressed as a fraction, for example Dynamic image 16 or Dynamic image 17 these numbers do not end or have a recurring pattern.

Surds

Surds are expressions for irrational numbers, for example Dynamic image 17

Simplifying surds

To simplify a surd, such as Dynamic image 19 firstly write it as the product of other integers, in this case Dynamic image 20. This can be expressed as Dynamic image 21. You know the square root of 9 is 3 and hence can simplify the expression further: Dynamic image 22

Examples

Simplify the following surds:
1. Dynamic image 23
2. Dynamic image 24
3. Dynamic image 25
4. Dynamic image 26

Here are the following worked solutions:
1. Dynamic image 23 = Dynamic image 28 = Dynamic image 29 = Dynamic image 30

2. Dynamic image 24 = Dynamic image 32 = Dynamic image 33 = Dynamic image 34

3. Dynamic image 25 = Dynamic image 36 = Dynamic image 37 = Dynamic image 38 = Dynamic image 39
4. Dynamic image 26 = Dynamic image 26

Rationalising the denominator

Knowing (x + y)(x – y) = x2 – y2 can be used to solve more complex surds

For example:

Simplify the following surd: Dynamic image 42

By multiplying both sides by Dynamic image 43 you can rationalise the denominator (i.e. turn it into a rational number):

Dynamic image 44

Using the rule mentioned above:

Dynamic image 45

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

Dynamic image 46

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