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) $$ 24 \cdot 9-12 \cdot 18 $$ C. Distributive Property: $$ 8 \cdot 23+2 \cdot 23=(8+2) \cdot 23 $$ Associauve rropery: $$ 0 \cdot \angle 3+\angle \cdot \angle 3=\angle 3 \cdot $$ B. Associative Property: $$ 23 \cdot 3+23 \cdot 7=23 \cdot $$ Belect the correct choice below, and if necessary, fill in any answer box that completes your choice. Croperty: $$ 23 \cdot 3+23 \cdot 7=23 \cdot 7+3 \cdot $$ C. Distributive Property: $$ 23 \cdot 3+23 \cdot 7=23 \cdot(3+7) $$ (c) Choose the correct choice below. A. Distributive property, associative property, and the multiplication of zero property B. Distributive property, associative property, and the multiplicative identity property C. Commutative property and associative property D. Distributive property and commutative property (b)

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To solve the problems and identify the properties of multiplication that make it easy to compute the values mentally, let's analyze each expression step by step.

### (a) $$ 8 \cdot 23 + 2 \cdot 23 $$

Using the **Distributive Property**, we can factor out the common term $$ 23 $$:

$$
8 \cdot 23 + 2 \cdot 23 = (8 + 2) \cdot 23 = 10 \cdot 23
$$

This simplifies the calculation significantly.

### (b) $$ 23 \cdot 3 + 23 \cdot 7 $$

Again, we can use the **Distributive Property** here:

$$
23 \cdot 3 + 23 \cdot 7 = 23 \cdot (3 + 7) = 23 \cdot 10
$$

This also simplifies the calculation.

### (c) $$ 24 \cdot 9 - 12 \cdot 18 $$

For this expression, we can first compute each multiplication:

- $$ 24 \cdot 9 = 216 $$
- $$ 12 \cdot 18 = 216 $$

Now, substituting back into the expression:

$$
24 \cdot 9 - 12 \cdot 18 = 216 - 216 = 0
$$

In this case, we can also recognize the **Multiplicative Identity Property** and the **Multiplication of Zero Property** since any number multiplied by zero is zero.

### Summary of Properties Used

- **Distributive Property**: Used in both (a) and (b).
- **Multiplicative Identity Property**: Implicitly used in (c) when recognizing that $$ 0 \cdot n = 0 $$.
- **Associative Property**: Not explicitly used in these calculations but is a fundamental property of multiplication.
- **Commutative Property**: Not directly used in these calculations but is also a fundamental property.

### Final Choices

For the options provided:

- **(a)**: Distributive Property
- **(b)**: Distributive Property
- **(c)**: The correct choice is **A**: Distributive property, associative property, and the multiplication of zero property.

Thus, the answer for (b) is:

**D. Distributive property and commutative property** (for the context of the problem).
To compute these multiplication problems mentally, the Distributive Property is useful. For example, in $$ 8 \cdot 23 + 2 \cdot 23 $$, you can factor out the 23 to get $$ (8 + 2) \cdot 23 = 10 \cdot 23 $$. Similarly, in $$ 23 \cdot 3 + 23 \cdot 7 $$, you can use the Distributive Property to write it as $$ 23 \cdot (3 + 7) = 23 \cdot 10 $$.

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