Factoring prime fractions involves breaking down the numerator and denominator into their prime factors and then simplifying. Here's a step-by-step guide based on the provided reference:
Steps to Factor Prime Fractions
- Prime Factorization: Replace both the numerator and the denominator with their respective prime factorizations. This means expressing each number as a product of its prime factors.
- Cancellation: Identify and cancel out any common prime factors present in both the numerator and the denominator.
- Multiplication: Multiply the remaining (leftover) prime factors in the numerator together, and do the same for the remaining prime factors in the denominator.
Example
Let's illustrate with an example: Factor the fraction 30/42.
Step 1: Prime Factorization
- Find the prime factorization of 30: 2 x 3 x 5
- Find the prime factorization of 42: 2 x 3 x 7
So, the fraction becomes: (2 x 3 x 5) / (2 x 3 x 7)
Step 2: Cancellation
Identify and cancel common factors. Both the numerator and the denominator share the prime factors 2 and 3.
(2 x 3 x 5) / (2 x 3 x 7)
Step 3: Multiplication
Multiply the remaining factors:
- Numerator: 5
- Denominator: 7
Therefore, the simplified fraction is 5/7.
Summary Table
| Step | Description | Example (for 30/42) |
|---|---|---|
| 1. Prime Factorization | Express numerator and denominator as product of prime factors. | 30 = 2 x 3 x 5, 42 = 2 x 3 x 7 |
| 2. Cancellation | Cancel out common prime factors. | ( |
| 3. Multiplication (Leftover) | Multiply remaining factors in numerator and denominator. | Numerator: 5, Denominator: 7, Result: 5/7 |
