Rewrite 8 + 12 using the GCF and factoring.
The Correct Answer and Explanation is :
To rewrite the expression (8 + 12) using the greatest common factor (GCF) and factoring, we first need to determine the GCF of the two numbers involved.
Step 1: Find the GCF
The numbers 8 and 12 can be broken down into their prime factors:
- (8 = 2^3)
- (12 = 2^2 \times 3)
Now, we identify the common prime factors. The prime factor common to both numbers is (2). The GCF is the product of the lowest powers of all common prime factors. In this case, the lowest power of (2) that appears in both factorizations is (2^2) (which equals 4).
Step 2: Factor Out the GCF
Now that we have determined the GCF to be (4), we can factor this out from the original expression:
[
8 + 12 = 4 \times 2 + 4 \times 3
]
This simplifies to:
[
8 + 12 = 4(2 + 3)
]
Step 3: Simplify the Factored Expression
Now we can simplify the expression inside the parentheses:
[
2 + 3 = 5
]
Thus, the factored form of the original expression becomes:
[
8 + 12 = 4 \times 5
]
Conclusion
In conclusion, we rewrote (8 + 12) using its GCF and factoring. We found that the GCF of 8 and 12 is 4, which allowed us to factor the expression into (4(2 + 3)). This not only simplifies the original addition but also provides insight into how different mathematical concepts like GCF and factoring interrelate. Factoring can be a powerful tool in simplifying expressions, solving equations, and understanding the relationships between numbers.