Which expression represents the associative property?

7. Step 1: Understand the associative property of multiplication. The associative property states that for any three numbers $a, b, c$, the way in which numbers are grouped in a multiplication operation does not change the product: $(a \times b) \times c = a \times (b \times c)$. Step 2: Apply the property to the given expression $(-7 \times 11) \times 4$. Here, $a = -7$, $b = 11$, and $c = 4$. According to the associative property, this is equivalent to $-7 \times (11 \times 4)$. Step 3: Compare this with the given options. Option A shows the distributive property. Option B changes the numbers being multiplied. Option C, $-7 \times (11 \times 4)$, correctly represents the associative property. Option D shows the commutative property.

The correct option is C.

8. Step 1: Simplify the expressions inside the parentheses. $(-2+6) = 4$ $(5-4+1) = (1+1) = 2$ Step 2: Substitute these simplified values back into the original expression. $3(4) - 2(2)$ Step 3: Perform the multiplications. $12 - 4$ Step 4: Perform the subtraction. $12 - 4 = 8$

The correct option is B.

9. Step 1: Find the additive inverse of $-2$. The additive inverse of a number $x$ is $-x$, such that $x + (-x) = 0$. For $-2$, the additive inverse is $-(-2) = 2$. Step 2: Find the multiplicative inverse of $-2$. The multiplicative inverse (reciprocal) of a non-zero number $x$ is $\frac{1}{x}$, such that $x \times \frac{1}{x} = 1$. For $-2$, the multiplicative inverse is $\frac{1}{-2} = -\frac{1}{2}$. Step 3: Combine the results. The additive inverse is $2$ and the multiplicative inverse is $-\frac{1}{2}$.

The correct option is D.

10. Step 1: Simplify the expression in the numerator. The expression is $2(-3) - (5) - 4(6+8)$. First, simplify inside the parentheses: $(6+8) = 14$. Substitute this back: $2(-3) - 5 - 4(14)$. Perform multiplications: $2(-3) = -6$ and $4(14) = 56$. The numerator becomes: $-6 - 5 - 56$. Combine the terms: $-11 - 56 = -67$. Step 2: Simplify the expression in the denominator. The expression is $5-6$. $5-6 = -1$ Step 3: Divide the numerator by the denominator. $\frac{-67}{-1} = 67$ Note: The calculated answer is $67$, which is not among the given options. Assuming there might be a typo in the question and that $(6+8)$ was intended to be $(6-8)$, let's re-evaluate the numerator: If the numerator was $2(-3) - (5) - 4(6-8)$: $2(-3) - 5 - 4(-2)$ $-6 - 5 + 8$ $-11 + 8 = -3$ In this case, the fraction would be $\frac{-3}{-1} = 3$. This matches option D. Given the multiple-choice format, we will proceed with this assumption to select an option.

The correct option is D.

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