Prime factorisation is the process of expressing a number as the product of its prime factors. To express this in exponential form, we first need to identify the prime factors of the number and then write them using exponents to indicate how many times each prime factor is used.
For example, let’s consider the number 60. The prime factorisation of 60 can be determined as follows:
- Start by dividing 60 by the smallest prime number, which is 2:
- 60 ÷ 2 = 30
- Continuing with 30, divide by 2 again:
- 30 ÷ 2 = 15
- 15 is not divisible by 2, so we move to the next prime number, which is 3:
- 15 ÷ 3 = 5
- 5 is a prime number itself, so we can stop here.
The prime factors of 60 are 2, 2, 3, and 5. In exponential form, we can express this as:
60 = 2² × 3¹ × 5¹
This shows that the number 2 is used twice, while both 3 and 5 are used once.
As another example, consider the number 72:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- Next, switch to 3:
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
The prime factorisation of 72 is 2, 2, 2, 3, and 3. In exponential form, it is:
72 = 2³ × 3²
In summary, expressing a number’s prime factorisation in exponential form allows us to succinctly represent the number and highlight the significance of its constituent prime factors.