Let's identify which of the given expressions are equivalent to [tex]\(3(n+6)\)[/tex].
First, simplify the expression [tex]\(3(n+6)\)[/tex]:
[tex]\[
3(n+6) = 3n + 3 \cdot 6 = 3n + 18
\][/tex]
So, we need to find which of the given choices are equivalent to [tex]\(3n + 18\)[/tex].
### Evaluate Each Option:
1. Option (a) [tex]\(3n + 6\)[/tex]:
Simplified form is already [tex]\(3n + 6\)[/tex].
[tex]\[
3n + 6 \nrightarrow 3n + 18
\][/tex]
This is not equivalent to [tex]\(3n + 18\)[/tex].
2. Option (b) [tex]\(3n + 18\)[/tex]:
Simplified form is already [tex]\(3n + 18\)[/tex].
[tex]\[
3n + 18 = 3n + 18
\][/tex]
This is equivalent to [tex]\(3n + 18\)[/tex].
3. Option (c) [tex]\(2n + 2 + n + 4\)[/tex]:
Combine like terms:
[tex]\[
2n + 2 + n + 4 = 2n + n + 2 + 4 = 3n + 6
\][/tex]
[tex]\[
3n + 6 \nrightarrow 3n + 18
\][/tex]
This is not equivalent to [tex]\(3n + 18\)[/tex].
4. Option (d) [tex]\(2(n + 6) + n + 6\)[/tex]:
Distribute and combine like terms:
[tex]\[
2(n + 6) + n + 6 = 2n + 12 + n + 6 = 2n + n + 12 + 6 = 3n + 18
\][/tex]
[tex]\[
3n + 18 = 3n + 18
\][/tex]
This is equivalent to [tex]\(3n + 18\)[/tex].
5. Option (e) [tex]\(2(n + 6) + n\)[/tex]:
Distribute and combine like terms:
[tex]\[
2(n + 6) + n = 2n + 12 + n = 2n + n + 12 = 3n + 12
\][/tex]
[tex]\[
3n + 12 \nrightarrow 3n + 18
\][/tex]
This is not equivalent to [tex]\(3n + 18\)[/tex].
### Conclusion:
The expressions that are equivalent to [tex]\(3(n + 6)\)[/tex] are:
- (b) [tex]\(3n + 18\)[/tex]
- (d) [tex]\(2(n + 6) + n + 6\)[/tex]
Thus, the correct answers are:
- (b) [tex]\(3n + 18\)[/tex]
- (d) [tex]\(2(n + 6) + n + 6\)[/tex]
