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# Index Notation

Index notation involves representing and reading numbers in certain ways. The expression 34 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.

34 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)

=  103 +  0 x 102  +  2 x 101 +  4 x 100

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 105  +  2 x 104  +  3 x 103  +  4 x 102  +  5 x 101  +  6 x 10

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