1. **Stating the problem:** Simplify and analyze the expression $$\sqrt{\left(\frac{20.5}{10 - 5a}\right)^{-\frac{1}{8}}}$$ and determine the values of $a$ for which the expression is defined. 2. **Rewrite the expression:** The square root can be written as a power of $\frac{1}{2}$, so the expression becomes: $$\left(\frac{20.5}{10 - 5a}\right)^{-\frac{1}{8} \times \frac{1}{2}} = \left(\frac{20.5}{10 - 5a}\right)^{-\frac{1}{16}}$$ 3. **Simplify the negative exponent:** Recall that $x^{-n} = \frac{1}{x^n}$, so: $$\left(\frac{20.5}{10 - 5a}\right)^{-\frac{1}{16}} = \left(\frac{10 - 5a}{20.5}\right)^{\frac{1}{16}}$$ 4. **Domain restrictions:** For the expression to be defined: - The denominator $10 - 5a \neq 0$ to avoid division by zero, so: $$10 - 5a \neq 0 \implies a \neq 2$$ - The base inside the root must be positive because we are taking a real 16th root (even root) of the base: $$\frac{10 - 5a}{20.5} > 0$$ Since $20.5 > 0$, this inequality depends on the numerator: $$10 - 5a > 0 \implies 5a < 10 \implies a < 2$$ 5. **Final domain:** Combining the above: $$a < 2$$ 6. **Summary:** The expression simplifies to: $$\left(\frac{10 - 5a}{20.5}\right)^{\frac{1}{16}}$$ which is defined for all real $a$ such that $a < 2$. **Answer:** The values of $a$ for which the expression is defined are all real numbers less than 2, i.e., $$a < 2$$.