Topic 1: Indices

Rules of Indices

An indices 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”.

Fact

[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.

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;

a × a × a × a × a = ?

...tells us that ‘a’ has been multiplied by itself 5 times. We write this as;

a × a × a × a × a = a⁵

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;

a⁴ × a³ = ?

Tip!

[In 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;

a × a ×  a × a × a × a × a = a⁴⁺³ = a⁷

The total number of as in the expansion is equal to the sum of powers in the question. This proves that an aⁿ × a^m = a^n+m

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, Example; a⁵ to the power 2.

This expression simply means;

a⁵ × a⁵ = ?

...we know that the first rule tells us that we should add the indices power together for multiplication;

a⁵ × a⁵ = a¹⁰

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.

For example;

(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.

(4⁵)² = 4^5×2 = 4¹⁰

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 = a^nm

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.

For example;

4⁸ ÷ 4² = ?

Expand the expression first to observe what is going on.

4⁸ ÷ 4²  = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4

                4 × 4

We can now use cancelling to simplify the expressions.

4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 46

4 × 4

There has been two cancellations which proves that we only need to find the difference between the two powers as shown below.

4⁸ ÷ 4² = 4^8-2 = 4⁶

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:

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 by using the third indices rule.

Proof!

We know that any number divide by itself is equal to zero.

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;

a^m/aⁿ = 1       

Notice here that we can use the third indices rule of subtraction which states that;

a^m ÷ aⁿ = a^m-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 we get 1. if we replace m and n with a single number;

(12^m)/(12^m) = 12^(m-m) = 12⁰ = 1 ≡ (12^m)/(12^m)

...or...

(12ⁿ)/(12ⁿ) = 12^(n-n) = 12⁰ = 1 ≡ (12ⁿ)/(12ⁿ)

Tip!

Dividing a number by itself = 1

Fifth indices rule:

The following indices rule deal with negative and fractional powers.

Be careful here 2^-4 is not the same as 2⁴ and it should not be related in anyway. Look at the pattern below;

2³ = 8

2² = 4

2¹ = 2

2⁰ = 1

2^-1 = 0.5

2^-2 = 0.25 = ½

2^-3 = 0.125 = ¼

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.

2^-3 = 1/2³ = 1/8 

         

The general rule for negative powers is;

a^-n = 1/aⁿ

          

^{For example;}

 2^-5/3 = ?

        

The reciprocal of;

2/3 = 3/2

     

That must mean that;

2^-5/3 = 3⁵/2

        

Next we simply power the denominator and numerator separately;

3⁵/2⁵= 243/32

Sixth indices rule:

The indices rule shown above is known as fractional indices rule. This is the simpler version but 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;

a^½ × a^½ = a^½+½ = a¹ = a

The expression above implies that a^½ is the √a. That proves the above rule that;

a^½ = √a

It also proves that;

a⅓ = ∛a

because...

a⅓ × a⅓ × a⅓ = a¹