The problem consists of two parts. The first part asks to identify the equation that demonstrates the associative property of multiplication. The second part asks to identify the equation that demonstrates the commutative property of addition.

1. Problem Description

The problem consists of two parts. The first part asks to identify the equation that demonstrates the associative property of multiplication. The second part asks to identify the equation that demonstrates the commutative property of addition.

2. Solution Steps

Part i) Associative Property of Multiplication

The associative property of multiplication states that for any real numbers a, b, and c, the following equation holds:

$a \times (b \times c) = (a \times b) \times c$

a) $2 \times (\sqrt{3} \times \pi) = (2 \times \sqrt{3}) \times \pi$. This equation follows the form of the associative property of multiplication.

b) $2 \times (\sqrt{3} + \pi) = (\sqrt{3} + \pi) \times 2$. This equation demonstrates the commutative property of multiplication.

c) $2 \times (\sqrt{3} + \pi) = 2 \times \sqrt{3} + 2 \times \pi$. This equation demonstrates the distributive property of multiplication over addition.

d) $2 + (\sqrt{3} \times \pi) = (\sqrt{3} \times \pi) + 2$. This equation demonstrates the commutative property of addition.

Therefore, option a) demonstrates the associative property of multiplication.

Part ii) Commutative Property of Addition

The commutative property of addition states that for any real numbers a and b, the following equation holds:

$a + b = b + a$

a) $2 + (\sqrt{3} + 0) = (2 + \sqrt{3}) + 0$. This equation demonstrates the associative property of addition.

b) $2 + (\sqrt{3} + 0) = 2 + \sqrt{3}$. This equation shows that $\sqrt{3} + 0 = \sqrt{3}$.

c) $2 + (\sqrt{3} + 0) = 2 + (0 + \sqrt{3})$. This equation shows that $\sqrt{3} + 0 = 0 + \sqrt{3}$, demonstrating the commutative property of addition.

d) $2 + (\sqrt{3} \times 1) = 2 + \sqrt{3}$. This equation shows that $\sqrt{3} \times 1 = \sqrt{3}$.

Therefore, option c) demonstrates the commutative property of addition.

3. Final Answer

i) a) $2 \times (\sqrt{3} \times \pi) = (2 \times \sqrt{3}) \times \pi$

ii) c) $2 + (\sqrt{3} + 0) = 2 + (0 + \sqrt{3})$