Indices are also known as powers, exponents or sometimes even orders. Note that indices is plural and index is singular.

Note that in the following:

a is the **power/exponent** and x is the **base**. In English, when a letter is smaller and on the upper right side of the bigger letter, we call it a superscript. In Maths, it is often a power.

Bear in mind that when you see an expression such as , this is 2 lots of x cubed. This follows from BIDMAS where powers are applied before multiplication. Many students get this confused with cubing 2x which, of course, gives a different answer of .

Students may find it hard to perform tasks with indices at first, especially in an algebraic setting. If you find that you are struggling, take a step back, try doing the calculations with numbers first.

It can be shown that indices abide by the following rules in maths:

## The Laws of Indices

- – think of multiplying by . You can write it out in full as . Hence, the powers are added. Note that this is only true if the base is the same and should not be applied to by , for instance.
- – similar to the previous example, however, when you are dividing algebraic terms you should subtract the powers.
- – anything to the power of zero is 1. You can see this from the previous bullet point by choosing a and b to be the same number.
- – consider taking to the power of 3, i.e. multiplying by itself 3 times. We have . It follows that the powers are multipled.
- – this is easy to see if you consider and subtracting the powers, then writing it as a fraction: .
- – can be seen if you consider and so must be the square root of x. This is because something multiplied by itself made x. Taking multiplied by itself 3 times shows that and the same applies for other fractions. It follows from this rule that .

Click here to see this list on the Things to Remember page.

## Example 1

## Example 2

Alternatively, click here to find Questions by Topic and scroll down to **all past INDICES questions** to practice some more.

**Are you ready to test your Pure Maths knowledge? If so, visit our Practice Papers page and take StudyWell’s own Pure **Maths** tests.**