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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:


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:

(3X106) + (6X105) + (5X104) + (7X103) + (4X102) + (2X101) + (8X100)

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.

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Question 1/5

1. The index notation used to express the prime factors of 36 would be:-

23 x 23

22 x 33

34 x 3

22 x 32


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