To help Cesar identify the field axioms applied in each equation, let's analyze them step by step.

Equation 1:

$(9 \cdot 10) \cdot 14 = 9 \cdot (10 \cdot 14)$ This equation involves rearranging the parentheses, which suggests the Associative Property of Multiplication. This property states that the grouping of factors in multiplication does not affect the result: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

Equation 2:

$\frac{1}{3} y + \frac{5}{8} x = \frac{5}{8} x + \frac{1}{3} y$ Here, the terms are swapped on either side of the equation, indicating the Commutative Property of Addition. This property states that the order of addition does not affect the sum: $a + b = b + a$

Equation 3:

$-7(19 + 8.5) = (-7)(19) + (-7)(8.5)$ This equation demonstrates the Distributive Property, where a factor is distributed over a sum: $a(b + c) = ab + ac$

Now, let’s complete the statements based on the field axioms applied:

1. The Distributive Rule is best represented by Equation 3.
2. The Associative Rule is best represented by Equation 1.
3. The Commutative Rule is best represented by Equation 2.

Would you like more details or have any questions? Here are five related questions to explore further:

1. What is the difference between associative and commutative properties?
2. Can the distributive property apply to subtraction as well?
3. How would these axioms work with matrices instead of numbers?
4. What happens when we apply these properties to division or subtraction?
5. Can we apply these axioms to more complex expressions involving exponents?

Tip: Understanding these basic field axioms helps simplify complex equations and improves problem-solving efficiency.