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.
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
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 are expressions for irrational numbers, for example
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:
Simplify the following surds:
Here are the following worked solutions:
1. = = =
2. = = =
3. = = = =
Rationalising the denominator
Knowing (x + y)(x – y) = x2 – y2 can be used to solve more complex surds
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: