# 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 “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; a^{n} x a^{m} = a^{n+m}

Below is an example of where this rule applies.

#### Example:

It’s is important to leave the answer in indices, notice how the final answer in our example is 2^{11} and not the actual value. 2^{11} is much easier to remember than 2048. Leaving the answer in indices form is particularly helpful when working with very large indices. For example 9^{8} is much easier to remember than 43046721.

## Second indices rule

This rule of indices is also popular. Suppose we had to rise a^{5} 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 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.

### Example

Work out (4^{5})^{2}:

The power of two means that we want to have 4^{5} 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 (a^{n})^{m} = a^{nm}

## 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.

### Example

Work out:

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.

### Proof

We shall now prove that any number to the power of zero is 1. In this proof we shall present two theories.

[MISSING CONTENT]

## Fifth Indices Rule

Above is another indices rule. Be careful here 2^{-4} is not the same as 2^{4} 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;

### Example

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 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.

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.

Thanks for this, very helpful stuff

Well-loved. Like or Dislike: 54 9

HELP!!!! I need to know what rule five is for a test tomorrow!! It says in my book:

2(in powers) -3=1 over 2, to the power 3= 1 over 8??? can

someone explain please?????

Like or Dislike: 19 15

Sorry, I could not get to your comment in time. I will be updating this entry with more content very soon.

Like or Dislike: 12 20

it is kind of unclear of how the rules are divided and explained because i think they are more rules. so that was kind of unclear.

Like or Dislike: 14 15

I will be updating this entry to make certain sections clearer very soon.

Like or Dislike: 4 5

what is A raise to the power x or A raise to the power 2x .

Like or Dislike: 13 16

A very helpful explanantion – thank you.

In the last line of rule four (just over the rule five title) am I right in thinking that the first fraction has been (inadventently) turn over again?

Like or Dislike: 10 8

my mind is fully clear after seeing this. thankyou.

Like or Dislike: 10 6

Thanks for dis!!!!

Like or Dislike: 12 7

Really helpful. Cheers

Like or Dislike: 8 5

Hey I need help.

How do you solve 3 to the power of 100 times 4 to the power of 100?

Like or Dislike: 7 3

Hi is that

\[ 3^{100} × 4^{100} \]

Remember you cannot apply any rule here since the bases for both numbers is different.

Like or Dislike: 7 3

You can simplify it to (3×4)^100, i.e 12^100

Like or Dislike: 4 3

thanks this really helps when trying to revise

Like or Dislike: 5 3

Thanks…How does the rule apply when adding or subtracting the same bases but powers eg. X^a + X^b

Like or Dislike: 4 5

Hello Lamin,

As you have seen above you can’t apply any of the rules if it is addition.

Like or Dislike: 3 8

Because its brackets basic multiplication and division they ARRRREEE the 3 rules though.

Like or Dislike: 3 2

awesome

Like or Dislike: 2 3

quite good

Like or Dislike: 1 2

Age

Like or Dislike: 1 2

how to solve

10^n – 4^n / 5^n – 2^n

Like or Dislike: 3 2

THANK YOU

Like or Dislike: 2 2

thx

Like or Dislike: 1 2

this is very help full web but i cant find the quation’s like

a1 over 2 x a-3 over 4

Like or Dislike: 0 3

this is very help full web but i cant find the quation’s like this one

a1 over 2 x a-3 over 4

Like or Dislike: 0 2

Hi most of your notes i find easy to understand but below my teacher (i’m an adult learner)he as given us stuff to reduce to a single index i have brackets within in brackets and fractions inside that and powers i’m confused have you a page i can look at please?

Like or Dislike: 0 2

Hello Thomas,

Can you type some examples of the questions that your teacher wants you to work out. Brackets means multiply so you might have to expand the expressions first. Are they similar to this:

\[(x^2y^3) ÷ (x^2y^4) \]

\[ (4^2(4×4×4)) × ((4×4×4)4^5) \]

Remember indices rules only apply when the base for all the terms is the same to be able to reduce it to a single indexed number.

Like or Dislike: 1 1

Hi, I am having difficulty working out this question. Can anybody help ?

5×2/3=x-1/3

The answer is 1/5 but I am having trouble getting to that answer.

Like or Dislike: 0 2