Rules of Indices
An indice is a number with a power; for example am; a is called the base and m is the power. The power is also often referred to as the “index” or “exponent”.
Below is the indices rules that you should familiar with. Notice that the a is constant within each rule.
First indices rule
This is a very popular rule in indices. This is known as the multiplication rule. Powers can be used to indicate how many times a (base) has been multiplied by itself. For example;
…tells us that ‘a’ has been multiplied by itself 5 times. We write this as;
The power ’5′ is the number of times that a has been multiplied by itself. In the rule we can replace m and n with 4 and 3 respectively;
Since the base is the same in both terms the multiplication is just a continuation. To observe this expand the expression;
The total number of as in the expansion is equal to the sum of powers in the question.
The example below shows where this rule applies.
Second indices rule
This rule of indices is known as the power of a power. A number with a power can be raised to a power eg; a5 to the power 2.
This expression simply means;
…we know that the first rule tells us that we should add the indices power together for multiplication;
But note also that 5×2 is equal to 10. This suggests that if we have am raised to the power n we simply multiply the powers together to get the result amxn or simply amn, this is proof for the second rule. Below are some examples of how to use this rule.
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.
Third indices rule
This rule states that for the division of two powered numbers, the result is equal the base to the power of the difference between the two powers as shown in the following example.
We can now use cancelling to simplify the expressions.
There has been two cancellations which proves that we only need to find the difference between the two powers as shown below.
Remember to leave your answers in indices, rather than the actual value. This is useful when carrying out much large/complex calculations.
Fourth Indices Rule
This next rule is very obvious (well… to remember). It is not so obvious why any number to the power zero is 1. We shall prove the following rule below.
We shall now prove that any number to the power of zero is 1. In this proof we shall present two theories.
Fifth Indices Rule
The following indices rule deal with negative and fractional powers.
This is another indices rule. Be careful here 2-4 is not the same as 24 and it should not be related in anyway. 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;
The reciprocal of;
That must mean that;
Next we simply power the denominator and numerator separately as shown below;
Sixth Indices Rule
The indices rule shown above is known as a 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 expression above implies that a½ is the √a. That proves the above rule that;
It also proves that;
This forms the general indices rule for fractional powers;
Now we shall look at an expansion to this rule. The indices rule explained above is derived from the following rule in fact. We have not powered the final answer with 1 because x to the power of 1 is equal to x.
This is just the harder fractional indices rule.
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;
Below are all the indices rules that you’ll need to remember.