Question
Which expressions are equivalent to g + h + (j + k)? Check all that apply.
g + (h + j) + k
(g + h) + j k
(g + h) + j + k
g + (h j) + k
g h + j k
g (h + j) k
g + h (j + k)
g + (h + j) + k
(g + h) + j k
(g + h) + j + k
g + (h j) + k
g h + j k
g (h + j) k
g + h (j + k)
Answers
GPT-4o mini
To determine which expressions are equivalent to \( g + h + (j + k) \), we can rearrange and apply the associative and commutative properties of addition. The only operations to consider are addition and parentheses; multiplication (represented by juxtaposition) and other operations do not operate the same way.
1. **\( g + (h + j) + k \)**: This is equivalent due to the associative property of addition.
**Equivalent**
2. **\( (g + h) + j k \)**: This expression suggests multiplication of \( j \) and \( k \) which is not equivalent to the original expression.
**Not equivalent**
3. **\( (g + h) + j + k \)**: This is equivalent due to the associative property of addition.
**Equivalent**
4. **\( g + (h j) + k \)**: This suggests multiplication of \( h \) and \( j \) which is not equivalent.
**Not equivalent**
5. **\( g h + j k \)**: This expression suggests multiplication of \( g \) and \( h \) as well as \( j \) and \( k \), which is not equivalent.
**Not equivalent**
6. **\( g (h + j) k \)**: This suggests multiplication, and it is not equivalent to the original expression.
**Not equivalent**
7. **\( g + h (j + k) \)**: This suggests multiplication between \( h \) and the sum \( (j + k) \), which is not equivalent.
**Not equivalent**
The expressions that are equivalent to \( g + h + (j + k) \) are:
- \( g + (h + j) + k \)
- \( (g + h) + j + k \)
Thus, the final answer includes both of these expressions as equivalent to the original expression.
1. **\( g + (h + j) + k \)**: This is equivalent due to the associative property of addition.
**Equivalent**
2. **\( (g + h) + j k \)**: This expression suggests multiplication of \( j \) and \( k \) which is not equivalent to the original expression.
**Not equivalent**
3. **\( (g + h) + j + k \)**: This is equivalent due to the associative property of addition.
**Equivalent**
4. **\( g + (h j) + k \)**: This suggests multiplication of \( h \) and \( j \) which is not equivalent.
**Not equivalent**
5. **\( g h + j k \)**: This expression suggests multiplication of \( g \) and \( h \) as well as \( j \) and \( k \), which is not equivalent.
**Not equivalent**
6. **\( g (h + j) k \)**: This suggests multiplication, and it is not equivalent to the original expression.
**Not equivalent**
7. **\( g + h (j + k) \)**: This suggests multiplication between \( h \) and the sum \( (j + k) \), which is not equivalent.
**Not equivalent**
The expressions that are equivalent to \( g + h + (j + k) \) are:
- \( g + (h + j) + k \)
- \( (g + h) + j + k \)
Thus, the final answer includes both of these expressions as equivalent to the original expression.