We know that:
5 × 5 = 25
The product 5 × 5 can be written as 5^{2}.
5 × 5 is known as the **expanded form** (or **factor form**) of 25 and 5^{2} 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**).
- 2
^{3} is read as '2 to the power 3' or simply '2 cubed'.
That is:
Example 1
Write 4^{3} as a number.
##### Solution:
Note the following:
- 64 = 4
^{3}
- 3 is the power (or index or exponent)
- 4 is the base number
- 64 is a basic numeral or number
- 4
^{3} is the index form (or power form) of 64
- 4 × 4 × 4 is the expanded form of 64
- For 64 = 4 × 4 × 4 = 4
^{3}, the base number 4
appears three times as a factor of the basic numeral (or number) 64
- 4
^{3} 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:
Key Terms
indices, expanded form, factor
form, index form, base, index, power, exponent, basic
numeral, basic number |