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To solve the given expression algebraically by applying exponentiation and root properties, we need to simplify each component and then sum the exponents of the variables.

### Expression:

\[
\left( \frac{n^8}{k^6} \right)^{-5} \cdot \frac{\sqrt[3]{\sqrt[3]{n^{180}}}}{\sqrt[5]{\sqrt[5]{k^{200}}}} \cdot \left( \frac{k^4}{n^6} \right)^{-6}
\]

### Step-by-step simplification:

1. **First term**:
   \[
   \left( \frac{n^8}{k^6} \right)^{-5} = \frac{(n^8)^{-5}}{(k^6)^{-5}} = \frac{n^{-40}}{k^{-30}} = n^{-40}k^{30}
   \]

2. **Second term**:
   \[
   \frac{\sqrt[3]{\sqrt[3]{n^{180}}}}{\sqrt[5]{\sqrt[5]{k^{200}}}} = \frac{\sqrt[9]{n^{180}}}{\sqrt[25]{k^{200}}} = \frac{n^{180/9}}{k^{200/25}} = \frac{n^{20}}{k^8}
   \]

3. **Third term**:
   \[
   \left( \frac{k^4}{n^6} \right)^{-6} = \frac{k^{-24}}{n^{-36}} = n^{36}k^{-24}
   \]

### Combine all terms:
\[
n^{-40}k^{30} \cdot \frac{n^{20}}{k^8} \cdot n^{36}k^{-24}
\]

Now, combine powers of \(n\) and \(k\):

- For \(n\):
  \[
  n^{-40} \cdot n^{20} \cdot n^{36} = n^{-40 + 20 + 36} = n^{16}
  \]
- For \(k\):
  \[
  k^{30} \cdot k^{-8} \cdot k^{-24} = k^{30 - 8 - 24} = k^{-2}
  \]

### Final expression:
\[
n^{16}k^{-2}
\]

The sum of the exponents of the variables \(n\) and \(k\) is:
\[
16 + (-2) = 14
\]

It seems like there was a possible mistake earlier in the available options of the sum, as 14 is not listed. However, based on this calculation, the sum should be **14**.

Would you like further clarification or more details on any specific part of this process? 

### Related Questions:
1. How do you simplify expressions involving negative exponents?
2. What is the general rule for multiplying terms with the same base?
3. How do you handle nested radicals when simplifying expressions?
4. Can you explain how exponent rules apply when dividing terms with the same base?
5. How would this expression change if the exponents were positive?

### Tip:
Always simplify nested radicals before applying exponent rules to avoid confusion and reduce errors in calculations.

Reducir la siguiente expresión algebraica aplicando propiedades de potenciación y radicación, e indicar la suma de los exponentes de las variables.

Learn how to simplify algebraic expressions by applying exponent and radical properties. This detailed solution involves step-by-step algebraic reduction and calculating the sum of the exponents. Suitable for Grades 11-12 students.

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