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 “the exponent”. Indices rules only apply when the base values are the same. This is very important to remember.
Below is the indices rules that you should familiar with.
First indices rule
This is a very popular rule in indices. We use powers to denote or indicate how many times something (the base) has been multiplied by itself over. For example;
…tells us that ‘a’ has been multiplied by itself 5 times. We write this as;
Suppose we had to work out an expression in the form;
We only need to add the powers because we’re continuing to multiply the bases. To see this we shall expand the expression like this;
So this proves that; an x am = an+m
Below is an example of where this rule applies.
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. 211 is much easier to remember than 2048. Leaving the answer in indices form is particularly helpful when working with very large indices. For example 98 is much easier to remember than 43046721.
Second indices rule
This rule of indices is also popular. Suppose we had to rise a5 to the power 2. The expression is;
This expression simply means;
…but we know that the first rule tells us that we should add the indices together to get;
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.
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 = anm
Third indices rule
As stated above indices rules only apply to numbers with the same base, above is another indices rule.
This is a worked out example of how to use the rule.
Let us expand the expression first to observe what is going on.
We can now use cancelling to simplify the expressions.
We get the answer 2. We can also use the rule above to check whether we get the same answer
We have managed to prove this rule.
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.
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
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;
Sixth Indices 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 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.
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.