To factor the expression [tex]\(48m + 42n\)[/tex] and identify the equivalent expressions, let's follow a detailed, step-by-step process:
1. Find the Greatest Common Divisor (GCD):
We need to find the greatest common divisor (GCD) of the coefficients [tex]\(48\)[/tex] and [tex]\(42\)[/tex].
- Prime factors of [tex]\(48\)[/tex]: [tex]\(48 = 2^4 \times 3\)[/tex]
- Prime factors of [tex]\(42\)[/tex]: [tex]\(42 = 2 \times 3 \times 7\)[/tex]
The common prime factors are [tex]\(2\)[/tex] and [tex]\(3\)[/tex]. So, the GCD is:
[tex]\[
2 \times 3 = 6
\][/tex]
2. Factor out the GCD:
We can factor [tex]\(6\)[/tex] out of the original expression:
[tex]\[
48m + 42n = 6(8m + 7n)
\][/tex]
3. Verify the Equivalent Expressions:
Next, we should check which of the given options are equivalent to [tex]\(6(8m + 7n)\)[/tex].
- Option A: [tex]\(6(8m + 7n)\)[/tex]
[tex]\[
6(8m + 7n) = 6(8m + 7n)
\][/tex]
This matches exactly with our factored expression.
- Option B: [tex]\(3(16m + 14n)\)[/tex]
[tex]\[
3(16m + 14n) = 3 \times 16m + 3 \times 14n = 48m + 42n
\][/tex]
Simplifies back to the original expression, which means it is also equivalent.
- Option C: [tex]\(2(12m + 21n)\)[/tex]
[tex]\[
2(12m + 21n) = 2 \times 12m + 2 \times 21n = 24m + 42n
\][/tex]
This is not equivalent to the original expression.
- Option D: [tex]\(7(7m + 6n)\)[/tex]
[tex]\[
7(7m + 6n) = 7 \times 7m + 7 \times 6n = 49m + 42n
\][/tex]
This is not equivalent to the original expression.
4. Conclusion:
The correct equivalent expressions for [tex]\(48m + 42n\)[/tex] are:
- [tex]\(6(8m + 7n)\)[/tex] (Option A)
- [tex]\(3(16m + 14n)\)[/tex] (Option B)
Therefore, the two correct answers are:
- A [tex]\(6(8m + 7n)\)[/tex]
- B [tex]\(3(16m + 14n)\)[/tex]
