The expression you are asked to evaluate is:

$\left( \frac{6^{-3} \cdot 6^5}{6^{-2}} \right)^{-\frac{1}{2}}$

Step-by-Step Solution:

1. Simplify the expression inside the parentheses: Using the properties of exponents, we can simplify the multiplication and division of terms with the same base $6$.

   • Combine the terms in the numerator: $6^{-3} \cdot 6^5 = 6^{-3 + 5} = 6^2$

   • Now simplify the entire fraction: $\frac{6^2}{6^{-2}} = 6^{2 - (-2)} = 6^{2 + 2} = 6^4$

2. Apply the outer exponent: Now we need to apply the outer exponent $-\frac{1}{2}$: $\left( 6^4 \right)^{-\frac{1}{2}} = 6^{4 \cdot -\frac{1}{2}} = 6^{-2}$

3. Final result: $6^{-2} = \frac{1}{6^2} = \frac{1}{36}$

So, the simplified expression evaluates to:

$\boxed{\frac{1}{36}}$

Let me know if you'd like further clarification or more details!

5 Related Questions:

1. How do you simplify expressions involving powers with the same base?
2. What is the rule for negative exponents and how does it apply in different contexts?
3. Can you explain why raising a number to a fractional power results in a root?
4. How does multiplying and dividing powers with the same base work?
5. How do you handle nested exponents or combinations of exponents?

Tip:

When simplifying expressions with exponents, always start by combining terms inside parentheses using the rules for exponents, then apply any outer powers at the end.