Question

Which of the pairs of expressions are equivalent?(1 point) Responses (k⋅18)⋅0.5 and k⋅(18⋅0.5) left parenthesis k times 18 right parenthesis times 0.5 and k times left parenthesis 18 times 0.5 right parenthesis x⋅(5+2) and (x⋅5)+2 x times left parenthesis 5 plus 2 right parenthesis and left parenthesis x times 5 right parenthesis plus 2 (d÷8)÷4 and d÷(8÷4) left parenthesis d divided by 8 right parenthesis divided by 4 and d divided by left parenthesis 8 divided by 4 right parenthesis (y−9)−8 and y−(9−8)

Answer 1

To determine which pairs of expressions are equivalent, we can analyze each pair:

1. **(k⋅18)⋅0.5 and k⋅(18⋅0.5)**:
- By the associative property of multiplication, both expressions evaluate to the same result: \( k \cdot 18 \cdot 0.5 = k \cdot (18 \cdot 0.5) \).
- **Equivalent**.

2. **(k times 18) times 0.5 and k times (18 times 0.5)**:
- Same as the first pair; both evaluate to the same result.
- **Equivalent**.

3. **x⋅(5+2) and (x⋅5)+2**:
- \( x \cdot (5 + 2) \) equals \( x \cdot 7 \).
- \( (x \cdot 5) + 2 \) does not equal \( x \cdot 7 \) unless \( x = 1 \).
- **Not Equivalent**.

4. **x times (5 plus 2) and (x times 5) plus 2**:
- Same reasoning as the previous pair; \( x \cdot (5 + 2) \) equals \( x \cdot 7 \) and \( (x \cdot 5) + 2 \) is not the same.
- **Not Equivalent**.

5. **(d÷8)÷4 and d÷(8÷4)**:
- \( (d ÷ 8) ÷ 4 \) is \( d ÷ (8 \cdot 4) = d ÷ 32 \).
- \( d ÷ (8 ÷ 4) \) is \( d ÷ 2 \).
- **Not Equivalent**.

6. **(y−9)−8 and y−(9−8)**:
- \( (y - 9) - 8 \) simplifies to \( y - 17 \).
- \( y - (9 - 8) \) simplifies to \( y - 1 \).
- **Not Equivalent**.

In conclusion, the equivalent pairs are:

- **(1) (k⋅18)⋅0.5 and k⋅(18⋅0.5)**
- **(2) (k times 18) times 0.5 and k times (18 times 0.5)**

The remaining pairs (3, 4, 5, and 6) are not equivalent.