As a term, the Law of Indices sounds intimidating. This is unfortunate, as indices are designed to simplify complex calculations to be more easily expressed. They help make expressions more concise.
In this guide, we will explore what indices are and the laws that allow us to use them. These laws include how to multiply, divide and use brackets with indices. If you need further advice, it may be worth finding a GCSE Maths tutor.
Index and Indices
The word "indices" is admittedly not very user-friendly. It is the plural of the word “indexâ€. Indices show how many times a number/letter has been multiplied by itself. For that reason, we have a base number and a small index attached to it.
The number below is written in index form:
53
- 5 is the base number
- 3 is the index
- This is a shortened way of writing 5 x 5 x 5
Index form can also be applied to a letter:
a5
- a is the base letter
- 5 is the index
- a has been multiplied by itself 5 times (a x a x a x a x a)
This is a preview of the versatility of indices. There are 6 rules or “laws†that govern the use of indices. These rules allow us to manipulate them in various ways.
An index can also be called a power. We will use the term index in this article.
1 - Multiply indices
When you multiply indices that have the same base number, you add the indices together:
23 x 22
- 2 is the shared base number
- 3 is the first index and 2 is the second index
- If we add the indices together, then 23 x 22 = 2 x 2 x 2 x 2 x 2. We can simplify this to 25
- You can also multiply indices with the same base letter e.g. a2 x a3 = a5
If you are asked a question regarding multiplying indices, it will normally ask you to “simplify†the calculation as shown above.
You can’t multiply indices when the base number or letter is not the same e.g. a2 x c3 can’t be multiplied.
1a
Simplify d5 x d2
1b
Simplify 6b5 x 4b-3
2 - Divide indices
In the same way that multiplying indices means adding the indices, dividing indices means subtracting them:
45 ÷ 42
- 4 is the shared base number
- 5 is the first index and 2 is the second index
- If we write the denominator and numerator in full, that gives us 4 x 4 x 4 x 4 x 4 / 4 x 4. We can cancel out these indices to be left with 4 x 4 x 4 x
4x4/4x4or 4 x 4 x 4, which can be simplified to 43 - The denominator and numerator are shown above so you can see the division clearly. In most circumstances, you can simply deduct the first index from the second index.
The following is another example, this time with a base letter:
a4/a2
= a4-2 = a22
Simplify 12f13 ÷ 4f7
3 - Brackets with indices
A base number with an index can be placed inside a bracket. An index will be outside the bracket, which means multiplying the indices together.
Take the following example:
(a3)2
There are several ways to simplify this expression:
- We can multiply everything by itself inside the bracket to the index outside the bracket e.g. (a3)2 = a3 x a3 = a x a x a x a x a x a = a6
- We can use the multiplication law of indices discussed earlier e.g. (a3)2 = a3 x a3 = a3+3 = a6
- The simplest method is to multiply the indices e.g. (a3)2 = a3x2 = a6
- As with all the laws, this can be used by base numbers and base letters
3
Write as a number to a single power: (92)3
4 - An index of zero
Sometimes an index will be 0, such as if we simplify an expression. When we multiply the base number/letter by itself zero times, this will return a value of one e.g. c0 = 1.
An index of zero can appear in several ways. One way is when using the division law of indices to simplify an expression, simplifying the indices to zero e.g. x2 ÷ x2 = x0.
The idea that multiplying by zero equals one is confusing, to put it mildly. It may help to remember that dividing anything by itself will always equal one e.g. 5 ÷ 5 = 1.
We can also change how we calculate indices by always beginning with a value of one. In the example below, we multiply 3 by itself 4 times:
34 = 3 x 3 x 3 x 3
If we add 1 to the beginning of this calculation, it won’t change the result. What we are saying in this example is to do 1 multiplied by 3 four times:
34 = 1 x 3 x 3 x 3 x 3
If we now work the value of the index down, eventually we will be left with just 1:
34 = 1 x 3 x 3 x 3 x 3
33 = 1 x 3 x 3 x 3
32 = 1 x 3 x 3
31 = 1 x 3
30 = 1
The final indices 30 is telling us to multiply 1 by 3 zero times, which leaves us with one.
4
Simplify g0
5 - Negative indices
When an index has a minus sign in front of it, this is called a negative index e.g. 3-2
We can create a negative index when dividing two terms with the same base number/letter:
s4 ÷ s5
There are two ways to simplify this example into a negative index:
- Dividing indices means subtracting them. The first index is smaller than the second, so it leads to a negative index e.g. s4 ÷ s5 = s4-5 = s-1
- We can use the numerator and denominator e.g. s x s x s x s / s x s x s x s x s. When we cancel the common factors, this leaves us with
sxsxsxs/sxsxsxsx s, which we can simplify to1/s. That means s4 ÷ s5 =1/s.
When we have simplified an expression, we need to make the negative index positive by putting it over 1 and reversing it. This is called finding the reciprocal:
e-2 = e-2/1 = 1/e2
In the example above, we have simplified the expression to e-2 and need to make it positive. First, we put it over 1, turning e-2 into e-2/1. Then we flip it and turn the index to positive: 1/e2.
The rule for negative indices described above is a-m = 1/am.
Negative indices are often used with other laws of indices, such as division and multiplication.
5
Simplify into index form: 3a-3
6 - Fractional indices
The following law is only required for Higher Mathematics GCSE.
A fraction index can be attached to a base number/letter. The example below shows a fractional index:
xa/b
- The denominator of the fraction (b) is the root of the base number/letter
- The numerator of the fraction (a) is the index of the base number/letter
The rule for a fractional index is am/n = (
n√a
)m.We will break this down step-by-step with the following example:
82/3
- The denominator for this fraction is 3. This means we need to find the cube root of 8. The cube root is 2 (2 x 2 x 2 = 8), which means 3√8= 2
- We know the ( n√a) part of the fractional index rule equals 2. The numerator is also 2, so we need to multiply these values together: (2)2 = 4
- The entire expression is 82/3 = ( 3√8)2 = (2)2 = 4
This calculation varies depending on the denominator. In the example above, the denominator was 3 we needed to find the cube root of the base number/letter. You can also encounter the following:
- A denominator of 2 means to find the square root e.g. 91/2 = = ±3
√9 - A denominator of 4 means to find the fourth root e.g. 161/4 = 4√16= ±2
6
Evaluate the following as an integer or fraction: 272/3
Glossary
- Common factor - A whole number that divides into another number exactly e.g. 3 is a common factor or 6, 9 and 12
- Denominator - The bottom part of a fraction e.g. 1/2
- Index - A number that is multiplied by itself one or more times. Another name for the index is power. The plural for the word index is indices.
- Numerator - The top part of a fraction e.g. 1/2
This post was updated on 01 Aug, 2023.
