Index notation involves representing and reading numbers in certain ways. The expression 3^{4} is read as ‘3 power of 4’. Here 3 is called the base, and 4 is called the index or exponent or power of the base.

3^{4} in index notation means 3 multiplied by itself 4 times, and is written as – 3 x 3 x 3 x 3.

When numbers have a power of 2, they are called squared, while power of 3 is called cubed.

Sometimes, we write a number in index notation with base 10. For example, 1024 can be written as –

1024 = 1000 + 0 + 20 + 4

1024 has 1 in its 1000th position, 0 in its hundredth position, 2 in its tenth position and 4 in its unit position. So

1024 = (10 x 10 x 10) + (0 x 10 x 10) + (2 x 10) + (4 x 1)

= 10^{3} + 0 x 10^{2} + 2 x 10^{1} + 4 x 10^{0}

Note that any number to the power of 0 is 1 (we will look at this in index notation in Algebra).

Expanded notation of a number can be given in two ways –

123456 = 100000 + 200000 + 30000 + 400 + 50 + 6

= 1 x 100000 + 2 x 100000 + 3 x 10000 + 4 x 100 + 5 x 10 + 6 x 1

In index form, this is written as –

123456 = 1 x 10^{5} + 2 x 10^{4} + 3 x 10^{3} + 4 x 10^{2} + 5 x 10^{1} + 6 x 1^{0}

In summary, to write a number in the expanded form:

- express the place value of each digit as the product of the digit and power of 10
- write the number as the sum of these products