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Let's solve the given expression step by step:

The expression to simplify is:

\[
\frac{1}{2x^2} + \frac{1}{\sqrt[5]{x^6}} - \frac{3}{x^{-6}}
\]

### Step 1: Simplify each term.
#### First term: \(\frac{1}{2x^2}\)

This term can be written as:
\[
\frac{1}{2x^2} = \frac{1}{2} \cdot x^{-2} = \frac{1}{2}x^{-2}
\]

#### Second term: \(\frac{1}{\sqrt[5]{x^6}}\)

This is a fractional exponent, so:
\[
\frac{1}{\sqrt[5]{x^6}} = \frac{1}{x^{6/5}} = x^{-6/5}
\]

#### Third term: \(\frac{3}{x^{-6}}\)

Applying the rules of exponents:
\[
\frac{3}{x^{-6}} = 3x^6
\]

### Step 2: Combine all terms.

Now, we can rewrite the original expression using these simplifications:
\[
\frac{1}{2}x^{-2} + x^{-6/5} - 3x^6
\]

### Step 3: Match with options.

Comparing this to the options given:

- The correct match is: \(\boxed{\frac{1}{2}x^{-2} + x^{-6/5} - 3x^6}\)

This corresponds to the fourth option.

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Would you like further clarification on any of the steps? Here are five related questions to deepen your understanding:

1. How do you handle fractional exponents when simplifying expressions?
2. Why does \(\frac{3}{x^{-6}}\) turn into \(3x^6\)?
3. Can you provide an example where exponents are negative in the denominator?
4. How does multiplying by fractions affect the power of variables?
5. What other ways can you rewrite expressions with roots and exponents?

**Tip:** Always simplify fractional exponents by converting roots into exponent form to make the calculation easier.

The expression 1/(2x^2) + 1/(x^(6/5)) - 3/(x^(-6)) can be written as:

Learn how to simplify the expression 1/(2x^2) + 1/(x^(6/5)) - 3/(x^(-6)) using exponent rules. This guide covers the key steps in transforming and simplifying complex rational expressions with exponents. Suitable for students in Grades 9-11.

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