# Index and expanded notation

## Index and expanded notation

Index notation is a short way of writing a number being multiplied by itself several times.

For example, instead of writing:

**4 ****x 4 ****x 4 **

It is easier to write:

**4 ^{3}**

The number that is being multiplied by itself is known as the **'base'**.

The number written above the base is known as the **'index'** or the **'power'**.

The index is the number of times that the base must be multiplied by itself.

## Factor trees

Factor trees are used to find the factors of a given number. Index notation can be used when a number is being expressed as a product of its prime factors.

For example:

## Place value and index notation

Indices can also be useful when writing large numbers. For example, each column of a value table can be expressed in powers of 10 by using index notation:

## Expanded notation

We can use the index notation above when writing numbers in expanded notation.

Writing a number in expanded notation means breaking that number up in relation to its value to the power of 10.

For example:

In expanded notation, the number 3 657 428 would be written as

3 X 1 000 000 + 6 X 100 000 + 5 X 10 000 + 7 X 1 000 + 4 X 100 + 2 X 10 + 8 X 1

Alternatively the number can be written using index notation:

(3X10^{6}) + (6X10^{5}) + (5X10^{4}) + (7X10^{3}) + (4X10^{2}) + (2X10^{1}) + (8X10^{0})

Usually, indexes of 1 and 0 are not included in index notation. This is because 101 is equal to 10 and 100 is equal to 1.