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 ...

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”.
Indices rules only apply when the base is the same for all terms. This is very important to remember.
Below is the indices rules that you should familiar with. Notice that the a is constant within each rule.
  • Rules Indices-02
  • Rules Indices-03
  • Rules Indices-04
  • Rules Indices-05
  • Rules Indices-06
  • Rules Indices-07

First indices rule

Rules Indices-01
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; Rules Indices-08 …tells us that ‘a’ has been multiplied by itself 5 times. We write this as; Rules Indices-09 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; Rules of Indices-05
In the multiplication rule we only need to add the powers.
Since the base is the same in both terms the multiplication is just a continuation. To observe this expand the expression; Rules of Indices-06 The total number of as in the expansion is equal to the sum of powers in the question.
This proves that; an x am = an+m
The example below shows where this rule applies.
Example: Workout 25 x 26
25 × 26 = 25+6 = 211
ExplanationWe must use the indices multiplication rule, we simply just add the powers. Rules of Indices-07
It’s is important to leave the answer in indices. 211 is much easier to remember than 2048.
Leaving the answer in indices form is particularly helpful when working with very large indices. 98 is much easier to remember than 43046721.
We can expand the expression to see exactly what is going on.
25 = 2 × 2 × 2 × 2 × 2


26 = 2 × 2 × 2 × 2 × 2 × 2

We can continue to multiply the two products together.

25 × 26 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 211


25 × 26 = 211

Second indices rule

Rules Indices-11
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. Rules Indices-10 This expression simply means; Rules of Indices-10 …we know that the first rule tells us that we should add the indices power together for multiplication; Rules of Indices-11 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.
Example: Work out (45)2:
Rules of Indices-12
Answer 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. Rules of Indices-13
This example proves the general rule of indices that (an)m = anm
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.

Third indices rule

Rules Indices-12
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.
Example: Work out 48 ÷ 42Rules of Indices-15
Answer: Expand the expression first to observe what is going on. Rules of Indices-16 We can now use cancelling to simplify the expressions. Rules of Indices-17 There has been two cancellations which proves that we only need to find the difference between the two powers as shown below. Rules of Indices-18
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

The fourth rule: a0 = 1
This next rule is very obvious (well… to remember). It is not so obvious why any number to the power zero is 1, but there is a number of ways this can be proved. In this article we shall prove it prove it by using the third indices rule.


We know that 2/2 = 1 the same is true to any number divide by it self;
4/4 = 1,     7/7 = 1,     40/40 = 1
The general idea is that a/a = 1. If this is true then that must also mean that;
am/an = 1
Notice here that we can use the indices rule of subtraction (third rule) which states that;
am ÷ an = am-n
We can replace m and n by the same number such that the number in the numerator is equal to the number in the denominator. To make sure that when we divide (equal numbers) we get 1. if we replace m and n with a single number;
(am)/(am)=a(m-m) = a(0) = 1 ≡ (am)/(am)


(an)/(an)=a(n-n) = a(0) = 1 ≡ (an)/(an)
Dividing a number by itself = 1
We can use actual numbers;
(a1)/(a1)=a(1-1) = a(0) = 1 ≡ (a1)/(a1)
(a2)/(a2)=a(2-2) = a(0) = 1 ≡ (a2)/(a2)
(a20)/(a20)=a(20-20) = a(0) = 1 ≡ (a20)/(a20)
(a100)/(a100)=a(100-100) = a(0) = 1 ≡ (a100)/(a100)

Fifth Indices Rule

The following indices rule deal with negative and fractional powers.
The fifth rule: a-1 = 1/a    and    a-m = 1/am
Be careful here 2-4 is not the same as 24 and it should not be related in anyway.
Look at the pattern below; Rules of Indices-21 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. Rules of Indices-22 If you’re to work it out on your calculator you might get one of these variations; Rules of Indices-23 The general rule for negative powers is; Rules of Indices-24
Example: Work out (2/3)-5;
Answer: We have to work out; Rules of Indices-25 The reciprocal of; Rules of Indices-26 That must mean that; Rules of Indices-27 Next we simply power the denominator and numerator separately as shown below; Rules of Indices-28

Sixth Indices Rule

Rules Indices-07
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; Rules of Indices-30 The expression above implies that a½ is the √a. That proves the above rule that; Rules of Indices-41 It also proves that; Rules of Indices-31 …because… Rules of Indices-32 This forms the general indices rule for fractional powers; Rules of Indices-33
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. Rules of Indices-34 This is just the harder fractional indices rule.
Example: Workout 272/3.
Answer: Rules of Indices-35 Here we could use the brackets rule by first breaking the power into parts. Rules of Indices-36 Thus we could write; Rules of Indices-37 Rules of Indices-38 Now we could use the rule for fractional powers. Rules of Indices-39
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;
Rules of Indices-40
Below are all the indices rules that you’ll need to remember.
Indices Rules