Let's go through each part of the question step-by-step:
### Factors of 48
The factors of 48 are the numbers that divide 48 exactly, without leaving a remainder. The complete list of factors for 48 is:
[tex]\[ \{ 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \} \][/tex]
### True or False: 27 is a perfect square.
To determine if 27 is a perfect square, we look for a number that, when squared, equals 27. The perfect squares close to 27 are:
[tex]\[ 5^2 = 25 \quad \text{and} \quad 6^2 = 36 \][/tex]
Since there is no integer whose square is 27, we conclude that:
[tex]\[ \text{False, 27 is not a perfect square.} \][/tex]
### Factor Completely
1. Factor [tex]\(7xy\)[/tex]:
[tex]\(7xy\)[/tex] is already in its simplest form as it is the product of the constants and the variables [tex]\(7\)[/tex], [tex]\(x\)[/tex], and [tex]\(y\)[/tex]. Thus:
[tex]\[ \boxed{7xy} \][/tex]
2. Factor [tex]\(2b^3 + 6b^2 + 4b\)[/tex]:
Factoring out the greatest common factor ([tex]\(2b\)[/tex]) from each term:
[tex]\[ 2b^3 + 6b^2 + 4b = 2b(b^2 + 3b + 2) \][/tex]
3. Factor [tex]\(m^2n^2 - 49\)[/tex]:
This is a difference of squares, which factors as:
[tex]\[ m^2n^2 - 49 = (mn)^2 - 7^2 = (mn - 7)(mn + 7) \][/tex]
4. Factor [tex]\(w^3 - 8\)[/tex]:
[tex]\(w^3 - 8\)[/tex] is a difference of cubes, which factors as:
[tex]\[ w^3 - 8 = w^3 - 2^3 = (w - 2)(w^2 + 2w + 4) \][/tex]
### Final Answers
- Factors of 48:
[tex]\[ \{ 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \} \][/tex]
- 27 is a perfect square:
[tex]\[ \text{False} \][/tex]
- Factor completely:
1. [tex]\( 7xy \)[/tex]
2. [tex]\( 2b(b^2 + 3b + 2) \)[/tex]
3. [tex]\( (mn - 7)(mn + 7) \)[/tex]
4. [tex]\( (w - 2)(w^2 + 2w + 4) \)[/tex]
