The problem asks which of the given expressions is equivalent to:

$\frac{1}{2} \left( \frac{4}{3}x^2 - \frac{1}{3} \right)$

Step-by-step Solution:

1. Distribute $\frac{1}{2}$ across the terms inside the parentheses:

   $\frac{1}{2} \times \left( \frac{4}{3}x^2 \right) - \frac{1}{2} \times \left( \frac{1}{3} \right)$

   $= \frac{1 \times 4}{2 \times 3}x^2 - \frac{1 \times 1}{2 \times 3}$

2. Simplifying the fractions:

   $= \frac{4}{6}x^2 - \frac{1}{6}$

   Which further simplifies to:

   $= \frac{2}{3}x^2 - \frac{1}{6}$

Now, let's compare this to the answer choices:

• A: $\frac{8}{3}(x^2 - \frac{2}{3})$
• B: $\frac{1}{3}(2x^2 - \frac{1}{2})$
• C: $\frac{1}{6}(x^2 - 1)$
• D: $\frac{2}{3}x^2 - \frac{2}{3}$

Checking each option:

• A and B do not match the simplified expression.

• C: When expanded, gives:

  $\frac{1}{6}x^2 - \frac{1}{6}$

  This matches the simplified form $\frac{2}{3}x^2 - \frac{1}{6}$.

So, the correct answer is C.

Would you like any further details or clarifications on this solution?

Here are 5 additional related questions:

1. How do you factor out constants from an algebraic expression?
2. What is the process for distributing fractions over expressions in parentheses?
3. Can we simplify further when dealing with multiple fractions?
4. How does multiplying a fraction by a variable term affect its coefficient?
5. What are other methods for checking equivalent expressions?

Tip: When distributing a fraction over a sum, always distribute the fraction to each term individually before simplifying.