The question asks to simplify and calculate the result of the given expression:

$\left(a^{4/3} \times a^{-3/8} \times a^{1/6}\right)^8$

Step-by-Step Solution:

1. Combine the exponents inside the parentheses: When multiplying powers with the same base $a$, you add the exponents. Therefore:

   $a^{4/3} \times a^{-3/8} \times a^{1/6} = a^{\left(\frac{4}{3} + \left(-\frac{3}{8}\right) + \frac{1}{6}\right)}$

2. Find a common denominator for the fractions: The denominators are 3, 8, and 6. The least common denominator (LCD) is 24.

   Rewrite each fraction with a denominator of 24: $\frac{4}{3} = \frac{32}{24}, \quad -\frac{3}{8} = -\frac{9}{24}, \quad \frac{1}{6} = \frac{4}{24}.$

   Now, add the fractions: $\frac{32}{24} - \frac{9}{24} + \frac{4}{24} = \frac{32 - 9 + 4}{24} = \frac{27}{24}.$

   Simplify the fraction: $\frac{27}{24} = \frac{9}{8}.$

   So: $a^{4/3} \times a^{-3/8} \times a^{1/6} = a^{9/8}.$

3. Apply the outer exponent: The expression is raised to the power of 8. When raising a power to another power, you multiply the exponents: $\left(a^{9/8}\right)^8 = a^{\left(\frac{9}{8} \cdot 8\right)} = a^9.$

Final Answer:

$\boxed{a^9}$

Let me know if you need further clarification or detailed steps!

Here are 5 related questions to deepen your understanding:

1. How do you calculate the least common denominator of multiple fractions?
2. What are the rules of exponents for multiplication and division of powers?
3. How do you handle negative exponents in algebraic expressions?
4. What is the process to simplify nested exponentiation, such as $(a^{m/n})^p$?
5. Can you explain the difference between fractional exponents and radicals?

Tip: Always simplify fractions and expressions step by step to minimize errors!