Indices

Indices

We know that:
5 × 5 = 25The product 5 × 5 can be written as 52.

5 × 5 is known as the expanded form (or factor form) of 25 and 52 is known as the index form of 25.

Generally when a number is multiplied by itself any number of times, the expression is simplified by using the index notation.

Note:
  • 2 is called the base.
  • 3 is called the index or power (or exponent) because it indicates the power to which the base, 2, is raised.
  • 8 is the basic numeral (or number).
  • 23 is read as ‘2 to the power 3’ or simply ‘2 cubed’.
That is:

Example 1

Write 43 as a number.

Solution:

Note the following:
  • 64 = 43
  • 3 is the power (or index or exponent)
  • 4 is the base number
  • 64 is a basic numeral or number
  • 43 is the index form (or power form) of 64
  • 4 × 4 × 4 is the expanded form of 64
  • For 64 = 4 × 4 × 4 = 43, the base number 4 appears three times as a factor of the basic numeral (or number) 64
  • 43 is read as ‘4 to the power 3’ or simply ‘4 cubed’

Example 2

Write each of the following expanded forms in index form:

Solution:

Example 3

Write each of the following in expanded form:

Solution:

Example 4

Find the value of the following:

Solution:

Example 5

Write 16 in index form using base 2.

Solution:

Example 6

Write the following numbers as a product of prime factors:

Solution: