The given mathematical expression is:

$\left( 2^{\frac{1}{2}} \right)^{-6}$

Step-by-step solution:

1. Simplify the exponent: $2^{\frac{1}{2}} \text{ is the same as } \sqrt{2}$ So, we can rewrite the expression as: $\left( \sqrt{2} \right)^{-6}$

2. Deal with the negative exponent: The negative exponent means we take the reciprocal: $\left( \sqrt{2} \right)^{-6} = \frac{1}{\left( \sqrt{2} \right)^6}$

3. Exponent of a square root: We know that: $\left( \sqrt{2} \right)^6 = \left( 2^{\frac{1}{2}} \right)^6 = 2^{\frac{1}{2} \times 6} = 2^3$

4. Final computation: Now, we have: $\frac{1}{2^3} = \frac{1}{8}$

Therefore, the correct answer is:

$\boxed{D. \frac{1}{8}}$

Would you like more details on any part of this explanation?

Here are 5 related questions to further your understanding:

1. What is the general rule for simplifying expressions with fractional exponents?
2. How do negative exponents affect the base of an expression?
3. Can you explain how to compute higher powers of square roots?
4. What happens when you multiply fractional exponents with integers?
5. What is the significance of the reciprocal in terms of negative exponents?

Tip: When dealing with negative exponents, always think of them as taking the reciprocal of the base raised to the positive of that exponent.