Question

writing about determining equivalent expressions
explain how to use properties to prove ( d + (8d + 1) + 7 ) is equivalent to ( 8 + 9d ).

sample response: use the associative property to move the parentheses, and the expression becomes ( (d + 8d) + 1 + 7 ). combine like terms, and the expression becomes ( 9d + 8 ). use the commutative property to...

check all that you included in your response.

• ☐ use the associative property to move the parentheses, and the expression becomes ( (d + 8d) + 1 + 7 ).
• ☐ combine like terms, and the expression becomes ( 9d + 8 ).
• ☐ use the commutative property to change the order of the terms, and the expression becomes ( 8 + 9d ).

Explanation:

Step1: Apply Associative Property

The associative property of addition states that \((a + b) + c = a + (b + c)\). For the expression \(d+(8d + 1)+7\), we can use the associative property to group \(d\) with \(8d\) and \(1\) with \(7\) as \((d + 8d)+(1 + 7)\).

Step2: Combine Like Terms

First, combine \(d+8d\). Since \(d = 1d\), using the distributive property (or just combining coefficients), \(1d+8d=(1 + 8)d=9d\). Then combine \(1 + 7=8\).

Step3: Form the Final Expression

After combining like terms, we have \(9d+8\). Then, using the commutative property of addition (\(a + b=b + a\)), we can rewrite \(9d + 8\) as \(8+9d\).

Answer:

To prove \(d+(8d + 1)+7\) is equivalent to \(8 + 9d\):

1. Associative Property: Rearrange grouping: \((d + 8d)+(1 + 7)\).
2. Combine Like Terms: \(d+8d = 9d\) (distributive/coefficient addition) and \(1 + 7=8\), giving \(9d+8\).
3. Commutative Property: Rearrange \(9d + 8\) to \(8+9d\).

Thus, \(d+(8d + 1)+7=8 + 9d\) by associative, distributive (for combining like terms), and commutative properties.