Let me transcribe the mathematical problem from the image and solve it step by step.

The given expression is:

$\sqrt{2 - \sqrt{7 - \sqrt{9 - \sqrt{12 + 6\sqrt{4} - \sqrt{16} - (\sqrt{3})^8}}}}.$

Step-by-Step Simplification:

1. Simplify the innermost terms:

   • $(\sqrt{3})^8 = (\sqrt{3}^2)^4 = 3^4 = 81$,
   • $\sqrt{4} = 2$,
   • $6\sqrt{4} = 6 \times 2 = 12$,
   • $\sqrt{16} = 4$.

   Substitute these into the expression inside the square root: $12 + 6\sqrt{4} - \sqrt{16} - (\sqrt{3})^8 = 12 + 12 - 4 - 81 = 24 - 85 = -61.$

   The innermost term becomes $-61$.

2. Work backward:

   • Substitute $-61$ back: $9 - \sqrt{-61}.$ Since this contains the square root of a negative number ($-61$), the result is complex, and the evaluation enters the realm of complex numbers.

Key Observation:

Due to the presence of $\sqrt{-61}$, the overall expression is not defined in the real number system. If the question assumes real values, the expression may need clarification or correction.

Would you like me to explore the problem in the complex plane, clarify any assumptions, or check for possible mistakes in the original setup?

Related Questions:

1. How are square roots of negative numbers handled in the complex number system?
2. Can simplifying nested radicals like this always yield real numbers?
3. What is the significance of the domain in which a mathematical expression is defined?
4. How do we simplify powers and roots efficiently in expressions like $(\sqrt{3})^8$?
5. When should we reject invalid inputs due to domain restrictions?

Tip:

Always check the domain of the expression (real vs. complex numbers) to avoid undefined operations.