What properties of multiplication make it easy to compute these values mentally? \( \begin{array}{l}\text { (a) } 8 \cdot 23+2 \cdot 23 \\ \text { (b) } 23 \cdot 3+23 \cdot 7 \\ \text { (c) } 24 \cdot 9-12 \cdot 18 \\ \text { A. Commutative Property: } 8 \cdot 23+2 \cdot 23=23 \cdot 8+\square \cdot 2 \\ \text { B. Associative Property: } 8 \cdot 23+2 \cdot 23=23 \cdot \square \cdot 23 \\ \text { C. Distributive Property: } 8 \cdot 23+2 \cdot 23=(8+2) \cdot \square\end{array} \) A.

One of the most helpful properties for mental calculations is the Distributive Property, which states that \(a(b+c) = ab + ac\). This allows you to group terms before multiplying. In your examples, you can add \(8\) and \(2\) before multiplying by \(23\). So, \(8 \cdot 23 + 2 \cdot 23\) can be simplified to \((8 + 2) \cdot 23 = 10 \cdot 23\), making it easier to compute mentally! Another valuable property is the Commutative Property, which says that changing the order of multiplication doesn’t change the product. For instance, in \(8 \cdot 23 + 2 \cdot 23\), you can rearrange it as \(23 \cdot 8 + 23 \cdot 2\). This can sometimes make it easier to see a multiplication that you can do mentally, especially if you know \(23\) and \(2\) or \(8\) are easier for you to calculate!