# Rules of Indices

^{m};

**a**is called the base and

**m**is the power. The power is also often referred to as the “index” or “exponent”.

## First indices rule

^{n}x a

^{m}= a

^{n+m}

^{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 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.

^{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}.

### Add your example

Use the form below to add another example to this section. You will need to provide an example in form of a question. Followed by the answer and then a detailed explanation/walkthrough of the example and answer.

Thanks for this, very helpful stuff

Well-loved. Like or Dislike: 70 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?????

Hot debate. What do you think? 23 17

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

Like or Dislike: 15 23

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: 17 19

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

Like or Dislike: 6 7

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

Like or Dislike: 15 18

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: 14 8

my mind is fully clear after seeing this. thankyou.

Like or Dislike: 13 7

Thanks for dis!!!!

Like or Dislike: 14 9

Really helpful. Cheers

Like or Dislike: 10 7

Hey I need help.

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

Like or Dislike: 10 4

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: 11 4

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

Like or Dislike: 6 6

thanks this really helps when trying to revise

Like or Dislike: 8 4

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

Like or Dislike: 5 7

Hello Lamin,

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

Like or Dislike: 4 10

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

Like or Dislike: 5 3

awesome

Like or Dislike: 3 3

quite good

Like or Dislike: 2 3

how to solve

10^n – 4^n / 5^n – 2^n

Like or Dislike: 5 4

THANK YOU

Like or Dislike: 4 3

thx

Like or Dislike: 2 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: 3 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: 1 3

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: 1 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: 3 2

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: 2 2

Probably not a problem still but thought i would answer anyway. times both sides by three cancels the division on both sides, adding 1 to both sides cancels the -1 on the right side and then you have 11= x

Like or Dislike: 1 1

i like mathematics

Like or Dislike: 5 2

Hi when a – number multiplyed by a – number does the answer become a + number.

Like or Dislike: 0 2

Hi when a – number multiplied by a – number does the answer become a + number.

Like or Dislike: 2 0

Yes. A negative number times a negative number is always equal to a postive number. For example;

\[-2 \times -2 = 4\]

\[-4 \times -4 = 16\]

\[-1 \times -1 = 1\]

Like or Dislike: 0 0

this really helps when studying …..thank you

Like or Dislike: 0 0