The number 48, expressed as a product of its prime numbers using exponents, is 2⁴ × 3.
Understanding prime factorization is a fundamental concept in number theory. It involves breaking down a composite number into its unique prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).
The Process of Prime Factorization for 48
To find the prime factorization of 48, we systematically divide the number by the smallest possible prime numbers until the quotient is 1.
- Start with the smallest prime number, 2:
- 48 ÷ 2 = 24
- Continue dividing the quotient by 2 as long as it is an even number:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- When 2 no longer divides evenly, move to the next prime number, 3:
- 3 ÷ 3 = 1
Once the quotient is 1, all the divisors used are the prime factors of the original number.
Step-by-Step Breakdown
The following table illustrates the division process:
| Step | Number | Divisor | Quotient | Prime Factors Found So Far |
|---|---|---|---|---|
| 1 | 48 | 2 | 24 | 2 |
| 2 | 24 | 2 | 12 | 2, 2 |
| 3 | 12 | 2 | 6 | 2, 2, 2 |
| 4 | 6 | 2 | 3 | 2, 2, 2, 2 |
| 5 | 3 | 3 | 1 | 2, 2, 2, 2, 3 |
From this process, we can see that 48 can be written as the product of its prime factors:
- 48 = 2 × 2 × 2 × 2 × 3
Expressing in Exponential Form
To write this prime factorization in exponential form, we group the repeated prime factors.
- The prime factor 2 appears four times (2 × 2 × 2 × 2). In exponential notation, repeated multiplication is represented by a base and an exponent. So, 2 × 2 × 2 × 2 can be written as 2⁴.
- The prime factor 3 appears once. When a prime factor appears only once, it can be written as 3¹ or simply 3.
Therefore, the prime factorization of 48 in exponential form is:
2⁴ × 3
This method provides a concise and standardized way to represent the unique prime building blocks of any composite number. For more detailed information on prime factorization, you can refer to educational resources like Khan Academy's explanation of prime factorization.
