Rules of Indices

An indice is a number with a power; for example a^m a is called the base and m is the power. The power is also often referred to as the “the index” or “the exponent”. Indices rules only apply when the base values are the same.

First indices rule

This is a very popular rule in indices. This is an example of where this rule applies.

Work out:

So this proves that; aⁿ x a^m = a^n+m it’s is important to leave the answer in indices, notice how the final answer in our example is 2¹¹ and not the actual value.

Second indices rule

As stated above indices rules only apply to numbers with the same base, the above is another indices rule.
This is a worked out example of how to use the rule.

Work out:

Try to work it out on your calculator and you’ll see. Remember to leave your answers in indices, rather than the actual value. This is also useful when carrying out much large/complex calculations.

Third indices rule

The above is another popular rule or law when working with indices. Take for example;
Work out (4⁵)²:

The power of two means that we want to have 4⁵ multiplied by itself 2 times. In this case we simply just multiply the powers together.

The answer for the above question is actually a very large number of 1048576 so you can see why it is important that you leave your answers in indices form. This example proves the general rule of indices that (aⁿ)^m = a^nm

Complex combinations of the above indices rules

Before we explore other indices rules, I think it’s best that we take a quick summary of the rules above. Here is a harder combination of the above rules.

1. Work out: 4x² x 4x⁶.

You have to be very careful when working out this kind of question. The initial thought might be to just add up the powers but won’t be the final answer.

Step one: First multiply the coefficients in our case to get 16;

then use the indices rule to work work out the x;

…then put the found answers together, that is;

… we do add up the powers using the indices rule, but also don’t forget to work out the coefficients.

2. Work out: 10x⁵ / 2x³.

First put the equation in a simple form and the carryout cancelling to simplify it.

From the second rule we know that;

That gives us the answer as;

…and that would be your answer.

The following indices rule deal with negative and fractional powers.

Fourth Rule

Above is another indices rule. Be careful here 2^-4 is not the same as 2⁴ and it should not be related in anyway. Let’s look at the pattern below;



By looking at the worked out indices above do you note a pattern?
A negative power on any number creates a reciprocal of that number. You can learn more about reciprocals here.

If you’re to work it out on your calculator you might get one of these variations;

The general rule for negative powers is;

Suppose we had to work out;

The reciprocal of;

That must mean that;

Next we simply power the denominator and numerator separately as shown below;

Fifth Rule

The about is the fractional indices rule. This is the simpler version but is not different from the one shown below.
You must know that anything to the power 1 is itself. So the expression shown below must be true;

The above expression implies that a^½ is the √a. That proves the above rule that;

It also proves that;

…because…

This forms the general indices rule for fractional powers;

Sixth Rule

This is just the harder fractional indices rule. Suppose we wanted to work out.

Here we could use the brackets rule by first breaking the power into parts.

Thus we could write;


Now we could use the rule for fractional powers.



You can see how this rule and the above are quite similar. We can split up any fractional power into a root followed by a power as shown below;

Seventh Rule

This next rule is very obvious

Below are all the indices rules that you’ll need to remember.