## 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 "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;

The total number of as in the expansion is equal to the sum of powers in the question.

^{n}x a

^{m}= a

^{n+m}

The example below shows where this rule applies.

^{5}x 2

^{6}

^{5}× 2

^{6}= 2

^{5+6}= 2

^{11}

^{11}is much easier to remember than 2048.

^{8}is much easier to remember than 43046721.

^{5}= 2 × 2 × 2 × 2 × 2

...and...

^{6}= 2 × 2 × 2 × 2 × 2 × 2

We can continue to multiply the two products together.

^{5}× 2

^{6}= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2

^{11}

...therefore...

^{5}× 2

^{6}= 2

^{11}

## 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; a^{5}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 5x2 is equal to 10. This suggests that if we have a

^{m}raised to the power n we simply multiply the powers together to get the result a

^{mxn}or simply a

^{mn}, this is proof for the second rule. Below are some examples of how to use this rule.

^{5})

^{2}:

^{5}multiplied by itself 2 times. In this case we simply just multiply the powers together.

^{n})

^{m}= a

^{nm}

**1048576**so you can see why it is important that you leave your answers in indices form.

## 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.^{8}÷ 4

^{2}

## Fourth Indices Rule

^{0}= 1

### Proof

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*

^{m}/a

^{n}= 1

^{m}÷ a

^{n}= a

^{m-n}

^{m})/(a

^{m})=a

^{(m-m)}= a

^{(0)}= 1 ≡ (a

^{m})/(a

^{m})

...or...

^{n})/(a

^{n})=a

^{(n-n)}= a

^{(0)}= 1 ≡ (a

^{n})/(a

^{n})

^{1})/(a

^{1})=a

^{(1-1)}= a

^{(0)}= 1 ≡ (a

^{1})/(a

^{1})

^{2})/(a

^{2})=a

^{(2-2)}= a

^{(0)}= 1 ≡ (a

^{2})/(a

^{2})

^{20})/(a

^{20})=a

^{(20-20)}= a

^{(0)}= 1 ≡ (a

^{20})/(a

^{20})

^{100})/(a

^{100})=a

^{(100-100)}= a

^{(0)}= 1 ≡ (a

^{100})/(a

^{100})

## Fifth Indices Rule

The following indices rule deal with negative and fractional powers.^{-1}= 1/a and a

^{-m}= 1/a

^{m}

^{-4}is not the same as 2

^{4}and it should not be related in anyway.

^{-5};

## 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 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;

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.

^{2/3}.

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 "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;

The total number of as in the expansion is equal to the sum of powers in the question.

^{n}x a

^{m}= a

^{n+m}

The example below shows where this rule applies.

^{5}x 2

^{6}

^{5}× 2

^{6}= 2

^{5+6}= 2

^{11}

^{11}is much easier to remember than 2048.

^{8}is much easier to remember than 43046721.

^{5}= 2 × 2 × 2 × 2 × 2

...and...

^{6}= 2 × 2 × 2 × 2 × 2 × 2

We can continue to multiply the two products together.

^{5}× 2

^{6}= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2

^{11}

...therefore...

^{5}× 2

^{6}= 2

^{11}

## 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; a^{5}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 5x2 is equal to 10. This suggests that if we have a

^{m}raised to the power n we simply multiply the powers together to get the result a

^{mxn}or simply a

^{mn}, this is proof for the second rule. Below are some examples of how to use this rule.

^{5})

^{2}:

^{5}multiplied by itself 2 times. In this case we simply just multiply the powers together.

^{n})

^{m}= a

^{nm}

**1048576**so you can see why it is important that you leave your answers in indices form.

## 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.^{8}÷ 4

^{2}

## Fourth Indices Rule

^{0}= 1

### Proof

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*

^{m}/a

^{n}= 1

^{m}÷ a

^{n}= a

^{m-n}

^{m})/(a

^{m})=a

^{(m-m)}= a

^{(0)}= 1 ≡ (a

^{m})/(a

^{m})

...or...

^{n})/(a

^{n})=a

^{(n-n)}= a

^{(0)}= 1 ≡ (a

^{n})/(a

^{n})

^{1})/(a

^{1})=a

^{(1-1)}= a

^{(0)}= 1 ≡ (a

^{1})/(a

^{1})

^{2})/(a

^{2})=a

^{(2-2)}= a

^{(0)}= 1 ≡ (a

^{2})/(a

^{2})

^{20})/(a

^{20})=a

^{(20-20)}= a

^{(0)}= 1 ≡ (a

^{20})/(a

^{20})

^{100})/(a

^{100})=a

^{(100-100)}= a

^{(0)}= 1 ≡ (a

^{100})/(a

^{100})

## Fifth Indices Rule

The following indices rule deal with negative and fractional powers.^{-1}= 1/a and a

^{-m}= 1/a

^{m}

^{-4}is not the same as 2

^{4}and it should not be related in anyway.

^{-5};

## 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 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;

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.

^{2/3}.