Question

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

Explanation:

Step1: Identify the property

We use the associative property of addition, which states that \( (a + b) + c = a + (b + c) \), and the commutative property of addition, which states that \( a + b = b + a \), along with the distributive property (or combining like terms). First, let's look at the expression \( d+(8d + 1)+7 \).

Step2: Apply associative property

Using the associative property of addition, we can re - group the terms: \( (d + 8d)+(1 + 7) \). The associative property allows us to change the grouping of the addends without changing the sum.

Step3: Combine like terms

For the terms with \( d \): \( d+8d=(1 + 8)d = 9d \) (by the distributive property \( ad+bd=(a + b)d \)). For the constant terms: \( 1 + 7=8 \).

Step4: Form the new expression

Now we have \( 9d+8 \). Then, using the commutative property of addition (\( a + b=b + a \)), \( 9d + 8=8 + 9d \).

Answer:

To prove \( d+(8d + 1)+7\) is equivalent to \(8 + 9d\), we use the associative, commutative, and distributive (for combining like terms) properties:

1. Associative Property of Addition: Re - group \(d+(8d + 1)+7\) as \((d + 8d)+(1 + 7)\). This property states \((a + b)+c=a+(b + c)\), so we can change the grouping of addends.
2. Combine Like Terms:

• For the variable terms: Using the distributive property \(ad+bd=(a + b)d\), \(d+8d=(1 + 8)d = 9d\).
• For the constant terms: \(1+7 = 8\). So now we have \(9d + 8\).

1. Commutative Property of Addition: Since \(a + b=b + a\), \(9d+8 = 8+9d\).

Thus, \(d+(8d + 1)+7\) is equivalent to \(8 + 9d\) by using the associative, commutative, and distributive (for combining like terms) properties of addition.