This question tests algebraic expression manipulation involving exponent rules and simplification. The correct approach is to expand $\left(\frac{1}{2}x\right)^2$ using the power rule: when squaring a product, square each factor. Thus $\left(\frac{1}{2}x\right)^2 = \left(\frac{1}{2}\right)^2 \cdot x^2 = \frac{1}{4}x^2$. Now we evaluate: $\frac{1}{4}x^2 - \frac{1}{4}x^2 = 0$ for all real $x$. The correct answer is B because these identical terms cancel completely. Choice D ($\frac{1}{4}x^2$) represents just the first term without the subtraction. Choice A ($\frac{1}{2}x^2$) might result from incorrectly squaring only the $x$ and not the coefficient $\frac{1}{2}$.

This question tests algebraic expression manipulation by recognizing equivalent forms of the same expression. The correct approach is to expand $(x-3)(x+3)$ using the difference of squares pattern: $(x-3)(x+3) = x^2 - 9$. Now we can substitute this into the original expression: $x^2 - 9 - (x^2 - 9) = x^2 - 9 - x^2 + 9 = 0$. The correct answer is A because the two identical expressions cancel completely. Choice C ($-18$) might result from incorrectly thinking we get $-9 - 9 = -18$. Choice E ($x^2 - 9$) represents just the first part of the expression without considering the subtraction.

This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator as a difference of squares and cancel the common factor with the denominator. The numerator factors to (x - 3)(x + 3). Canceling the (x - 3) term with the denominator for x ≠ 3 yields x + 3. This matches the correct answer of x + 3. A representative distractor like x - 3 fails due to incorrectly factoring or subtracting instead of adding. Another error could involve not canceling and leaving it as $x^2$ - 9.

This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the 2/3 through the parentheses. Distribute to 9 to get 6, and to -3w to get -2w. The result is 6 - 2w. This matches the correct answer of 6 - 2w. A representative distractor like 6 + 2w fails due to a sign error in distribution. Another mistake could be misapplying the fraction, leading to 6 - w.

This question tests algebraic expression manipulation involving factoring and simplification of rational expressions. The correct simplification strategy is to recognize that the numerator $a^2 - b^2$ is a difference of squares that factors as $(a+b)(a-b)$. Substituting this factorization gives us $\frac{(a+b)(a-b)}{a-b}$. Since $a \neq b$, we know that $a-b \neq 0$, so we can cancel the common factor $(a-b)$ from numerator and denominator. This leaves us with $a+b$, which is choice B. Choice A ($a-b$) might result from incorrectly thinking the expression simplifies to the denominator. Choice D ($a^2-b^2$) represents the numerator alone without considering the division.

This question tests algebraic expression manipulation. The correct evaluation strategy is to substitute k = -3 into the expression and compute. Substitute to get $(-3)^2$ - 2(-3) + 1. Compute 9 + 6 + 1 = 16. This matches the correct answer of 16. A representative distractor like 4 fails due to a sign error in the linear term. Another mistake could be incorrect exponentiation, leading to 7.

This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the 5 and then combine like terms with the +4z. Distribute 5 to get 5 - 10z. Add 4z to get 5 - 10z + 4z = 5 - 6z. This matches the correct answer of 5 - 6z. A representative distractor like 5 + 6z fails due to a sign error in distribution. Another common mistake is incorrect combining, leading to 9 - 2z.

This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator and cancel the common factor. Factor (x + $1)^2$ - (x + 1) as (x + 1)(x + 1 - 1) = (x + 1)x. Cancel (x + 1) with the denominator for x ≠ -1, leaving x. This matches the correct answer of x. A representative distractor like x + 1 fails due to incomplete factoring. Another error is not subtracting correctly, leading to $x^2$ + 1.

This question tests algebraic expression manipulation. The correct simplification strategy is to find a common denominator and combine the fractions. The common denominator is 4, so rewrite as (2x / 4) - (3x / 4). Combine to (2x - 3x) / 4 = -x / 4. This matches the correct answer of -x / 4. A representative distractor like x / 4 fails due to a sign error in subtraction. Another mistake could be adding denominators, leading to something like 4x / 8.

This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator and cancel the common factor with the denominator. The numerator is (x - 2)(x + 2). Cancel (x + 2) with the denominator for x ≠ -2, leaving x - 2. This matches the correct answer of x - 2. A representative distractor like x + 2 fails due to a sign error in cancellation. Another mistake is not simplifying, leaving (x - 2)/(x + 2).