What properties of multiplication make it easy to compute these values mentally? (a) \( 8 \cdot 23+2 \cdot 23 \) (b) \( 23 \cdot 3+23 \cdot 7 \) (c) \( 24 \cdot 9-12 \cdot 18 \) (a) Select the correct choice below, and if necessary, fill in any answer box that completes your choice. A. Commutative Property: \( 8 \cdot 23+2 \cdot 23=23 \cdot 8+\quad \cdot 2 \) B. Associative Property: \( 8 \cdot 23+2 \cdot 23=23 \cdot \cdot 23 \) C. Distributive Property: \( 8 \cdot 23+2 \cdot 23=(8+2) \cdot 23 \) (b) Select the correct choice below, and if necessary, fill in any answer box that completes your choice. A. Associative Property: \( 23 \cdot 3+23 \cdot 7=23 \cdot \square \cdot 23 \) B. Commutative Property: \( 23 \cdot 3+23 \cdot 7=23 \cdot 7+3 \). \( \square \) C. Distributive Property: \( 23 \cdot 3+23 \cdot 7=\square \cdot(3+7) \)
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Multiplication has some fantastic properties that can really simplify calculations! For example, in part (a), the Distributive Property allows you to combine \(8 \cdot 23 + 2 \cdot 23\) into \((8 + 2) \cdot 23\), making mental math a breeze. In part (b), you can apply the Distributive Property again to rewrite \(23 \cdot 3 + 23 \cdot 7\) as \(23 \cdot (3 + 7)\), which equals \(23 \cdot 10\). When using these properties, it’s always beneficial to look for patterns or common factors. Many stumble by overlooking the fact that a common multiplier, like \(23\) in your examples, can simplify expressions greatly. Remember, grouping numbers strategically can transform complex problems into simple ones, making your mental math skills shine!
