Question

Complete the following stepped-out solution to show that −7+[(8⋅m)⋅1/8]+7
is equivalent to m
. (2 points)
−7+____+[(8⋅m)⋅1/8]
Commutative Property of Addition
___+[(8⋅m)⋅1/8]
Inverse Property of Addition
(8⋅m)⋅1/8
Identity Property of Addition
(8⋅___) ⋅ m
Associative Property of Multiplication
(___) ⋅m
Inverse Property of Multiplication
m
Identity Property of Multiplication

1 answer

To show that \( -7 + [(8 \cdot m) \cdot \frac{1}{8}] + 7 \) is equivalent to \( m \), we can fill in the blanks step-by-step based on the properties of addition and multiplication.

  1. Start with the original expression: \[ -7 + \underline{7} + [(8 \cdot m) \cdot \frac{1}{8}] \]

    • Here, we add \( 7 \) to balance out the \( -7 \), using the Inverse Property of Addition.
  2. Now we have: \[ 0 + [(8 \cdot m) \cdot \frac{1}{8}] \]

    • Here, we recognize that \( -7 + 7 = 0 \), following the Identity Property of Addition.
  3. Then, we notice: \[ [(8 \cdot m) \cdot \frac{1}{8}] \]

    • This expression is still intact and represents the product of \( 8 \cdot m \) and \( \frac{1}{8} \).
  4. Now we can simplify \( (8 \cdot m) \cdot \frac{1}{8} \) using the Associative Property of Multiplication: \[ (8 \cdot \underbrace{\frac{1}{8}}_{1}) \cdot m \]

    • The value \( 8 \cdot \frac{1}{8} \) equals \( 1 \).
  5. Thus, we have: \[ 1 \cdot m \]

    • This simplifies further using the Identity Property of Multiplication, where \( 1 \cdot m = m \).

Putting it all together, we fill in the blanks:

  • First blank: 7 (to move from \( -7 \) to \( 0 \))
  • Second blank: 0 (as \( -7 + 7 = 0 \))
  • Fourth blank: 1 (because \( (8 \cdot \frac{1}{8}) = 1 \))

The completed stepped-out solution looks like:

\[ -7 + 7 + [(8 \cdot m) \cdot \frac{1}{8}] \]

  1. Inverse Property of Addition: \[ 0 + [(8 \cdot m) \cdot \frac{1}{8}] \]
  2. Identity Property of Addition: \[ [(8 \cdot m) \cdot \frac{1}{8}] \]
  3. Associative Property of Multiplication: \[ (8 \cdot \frac{1}{8}) \cdot m \]
  4. Inverse Property of Multiplication: \[ 1 \cdot m \]
  5. Identity Property of Multiplication: \[ m \]

Therefore, the whole expression is equivalent to \( m \).