Lesson 3 Homeworl Practice Properties of Operations Name the property shown by each statement. \( \begin{array}{l}\text { 1. } 1 \cdot(a+3)=a+3 \\ \begin{array}{l}\text { 3. }(a b)_{c}=c(a b) \\ \text { 5. } m(n r)=(m n) r\end{array} \\ \text { 2. } 2 p+(3 q+2)=(2 p+3 q)+2 \\ \text { state whether the following conjectures are true or false. If false, } \\ \text { 7. The product of an odd number and an even number is always odd. } \\ \text { 8. The sum of two whole numbers is always larger than either whole number. } \\ \text { 4. } 2 t \cdot 0=0\end{array} \) Simplify each expression. Justify each step. 9. \( 2 d(3) \)

Let's go through the problems step by step. ### Naming the Properties of Operations 1. **Statement:** \( 1 \cdot (a + 3) = a + 3 \) **Property:** This is an example of the **Identity Property of Multiplication**. It states that multiplying any number by 1 will yield the same number. 2. **Statement:** \( 2p + (3q + 2) = (2p + 3q) + 2 \) **Property:** This is an example of the **Associative Property of Addition**. It states that the way in which numbers are grouped in addition does not change their sum. 3. **Statement:** \( (ab)c = c(ab) \) **Property:** This is an example of the **Commutative Property of Multiplication**. It states that the order in which numbers are multiplied does not change the product. 4. **Statement:** \( 2t \cdot 0 = 0 \) **Property:** This is an example of the **Multiplicative Property of Zero**. It states that any number multiplied by zero equals zero. 5. **Statement:** \( m(nr) = (mn)r \) **Property:** This is an example of the **Associative Property of Multiplication**. It states that the way in which numbers are grouped in multiplication does not change their product. ### True or False Conjectures 7. **Conjecture:** The product of an odd number and an even number is always odd. **Answer:** **False.** The product of an odd number and an even number is always even. For example, \(3 \cdot 2 = 6\) (even). 8. **Conjecture:** The sum of two whole numbers is always larger than either whole number. **Answer:** **False.** The sum of two whole numbers can be equal to one of the numbers if one of them is zero. For example, \(0 + 5 = 5\) (not larger than 5). ### Simplifying the Expression 9. **Expression:** \( 2d(3) \) To simplify this expression, we can multiply the constants together: \[ 2d(3) = 2 \cdot 3 \cdot d \] Now, calculate \(2 \cdot 3\): \[ 2 \cdot 3 = 6 \] Therefore, the simplified expression is: \[ 6d \] ### Summary of Answers 1. Identity Property of Multiplication 2. Associative Property of Addition 3. Commutative Property of Multiplication 4. Multiplicative Property of Zero 5. Associative Property of Multiplication 6. Conjecture 7: False 7. Conjecture 8: False 8. Simplified Expression: \( 6d \)