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.
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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.
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Now we have: \[ 0 + [(8 \cdot m) \cdot \frac{1}{8}] \]
- Here, we recognize that \( -7 + 7 = 0 \), following the Identity Property of Addition.
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Then, we notice: \[ [(8 \cdot m) \cdot \frac{1}{8}] \]
- This expression is still intact and represents the product of \( 8 \cdot m \) and \( \frac{1}{8} \).
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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 \( 1 \).
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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:
\[ -7 + 7 + [(8 \cdot m) \cdot \frac{1}{8}] \]
- Inverse Property of Addition: \[ 0 + [(8 \cdot m) \cdot \frac{1}{8}] \]
- Identity Property of Addition: \[ [(8 \cdot m) \cdot \frac{1}{8}] \]
- Associative Property of Multiplication: \[ (8 \cdot \frac{1}{8}) \cdot m \]
- Inverse Property of Multiplication: \[ 1 \cdot m \]
- Identity Property of Multiplication: \[ m \]
Therefore, the whole expression is equivalent to \( m \).
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