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:
 
In the following sections, we will consider the six index laws.
Index Law for Multiplication
We know that:
In general:
This formula tells us that when multiplying powers with the same base, add the indices.
This is the first index law and is known as the Index Law for Multiplication.
Example 7
Solution:
 We know that:
In general:
This formula tells us that when dividing powers with the same base, the index in the denominator is subtracted from the index in the numerator.
This is the second index law and is known as the Index Law for Division.
Example 8
Solution:
Note:
Simplify the numerical coefficients first, and then apply the index law.
In general:
This formula tells us that any number, except 0, raised to the power zero has a numerical value of 1.
This is the third index law and is known as the Power of Zero.
Example 9
Solution:
 We know that:
In general:
This formula tells us that when a power of a number is raised to another power, multiply the indices.
This is the fourth index law and is known as the Index Law for Powers.
Example 10
Solution:
Note:
Always remove brackets first.
Example 11
Simplify each of the following:
Solution:
Index Law for Powers of Products
|
 We know that:
In general:
This formula tells us that when a product is raised to a power, every factor of the product is raised to the power.
This is the fifth index law and is known as the Index Law for Powers of Products.
Example 12
Simplify each of the following:
Solution:
Example 13
Solution:
Index Law for Powers of Quotients
|
 We know that:
In general:
This formula tells us that when a quotient is raised to a power, both the numerator and denominator are raised to the power.
This is the sixth index law and is known as the Index Law for Powers of Quotients.
Example 14
Simplify each of the following:
Solution:
Example 15
Solution:
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