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.