Let's solve each step-by-step and match with the correct property. --- ## 1. \( 7 \times 2 \times (4 \times 1) = 7 \times 2 \times 4 \times 1 \) ### Property: **Associative property of multiplication** **Explanation:** The associative property allows you to regroup numbers when multiplying without changing the result. \[(a \times b) \times c = a \times (b \times c)\] --- ## 2. \( 9 \times (4 \times 12) = (9 \times 4) \times 12 \) ### Property: **Associative property of multiplication** **Explanation:** This is also the associative property of multiplication, as the grouping of the numbers changes, not their order. --- ## 3. \( \left(\frac{1}{2} + 2\right) = 5 + 4 + \) ### Property: **Additive inverse** (if you meant \(2 + (-2) = \)), otherwise this looks incorrect. **Explanation:** If the intention was to show that a number plus its additive inverse equals zero. --- ## 4. \( (4 + 19) + 7 = (19 + 4) + 7 \) ### Property: **Commutative property of addition** **Explanation:** Changing the order of addends does not change the sum. \[(a + b) = (b + a)\] --- ## 5. \( (13 \times 41) \times 6 = (41 \times 13) \times 6 \) ### Property: **Commutative property of multiplication** **Explanation:** Changing the order of factors does not change the product. \[(a \times b) = (b \times a)\] --- ## 6. \( 6 \times (x + 3) = 3 \times (10 + y) \) ### Property: **Not matching any standard property** This equation doesn't fit a standard property as written. If intended as \(6 \times (x + 3) = 6x + 18\), that would be distributive property. --- ## 7. \( 6(x + 3) = 6x + 18 \) ### Property: **Distributive property of multiplication over addition** **Explanation:** Multiplying a number by a sum is the same as multiplying each addend separately and adding the products. \[a(b + c) = ab + ac\] --- ## 8. \( (5p - 2)6 = 30p - 12 \) ### Property: **Distributive property of multiplication over subtraction** **Explanation:** Multiplying a number by a difference is the same as multiplying each term and subtracting the products. \[a(b - c) = ab - ac\] --- ## **Summary Table** | # | Expression | Property | |---|------------------------------------------------|-----------------------------------------------| | 1 | \( 7 \times 2 \times (4 \times 1) = 7 \times 2 \times 4 \times 1 \) | Associative property of multiplication | | 2 | \( 9 \times (4 \times 12) = (9 \times 4) \times 12 \) | Associative property of multiplication | | 3 | \( \left(\frac{1}{2} + 2\right) = 5 + 4 + \) | Additive inverse (if corrected) | | 4 | \( (4 + 19) + 7 = (19 + 4) + 7 \) | Commutative property of addition | | 5 | \( (13 \times 41) \times 6 = (41 \times 13) \times 6 \) | Commutative property of multiplication | | 6 | \( 6 \times (x + 3) = 3 \times (10 + y) \) | Not a standard property as written | | 7 | \( 6(x + 3) = 6x + 18 \) | Distributive property | | 8 | \( (5p - 2)6 = 30p - 12 \) | Distributive property | --- If you need clarification or if there is a typo in question 3 or 6, please let me know!