Let's solve the expression step by step.

Given the expression:

$\frac{(16^{-2.5})^{-0.2}}{16^{3/4}}$

Step 1: Simplify $(16^{-2.5})^{-0.2}$

Use the rule of exponents that $(a^m)^n = a^{m \cdot n}$.

$(16^{-2.5})^{-0.2} = 16^{-2.5 \cdot (-0.2)} = 16^{0.5}$

So, $(16^{-2.5})^{-0.2} = 16^{0.5}$.

Step 2: Substitute back into the original expression

Now the expression becomes:

$\frac{16^{0.5}}{16^{3/4}}$

Step 3: Apply the exponent division rule

When dividing exponents with the same base, you subtract the exponents:

$\frac{16^{0.5}}{16^{3/4}} = 16^{0.5 - 3/4}$

First, find $0.5 - 3/4$:

$0.5 = \frac{1}{2}, \quad \frac{1}{2} - \frac{3}{4} = \frac{2}{4} - \frac{3}{4} = -\frac{1}{4}$

Thus:

$\frac{16^{0.5}}{16^{3/4}} = 16^{-\frac{1}{4}}$

Step 4: Simplify the result

We now know that $16 = 2^4$, so:

$16^{-\frac{1}{4}} = (2^4)^{-\frac{1}{4}} = 2^{-1} = \frac{1}{2}$

Final Answer:

$\frac{(16^{-2.5})^{-0.2}}{16^{3/4}} = \frac{1}{2}$

Let me know if you need more details, or if you'd like further clarification! Here are some follow-up questions to deepen your understanding:

1. What is the general rule for dividing exponents with the same base?
2. How can we apply the properties of exponents to simplify more complex expressions?
3. How do negative exponents affect the overall result of an expression?
4. What are some applications of fractional exponents in real-world problems?
5. How do you convert between different forms of exponential expressions (e.g., from base 16 to base 2)?

Tip: When working with fractional exponents, remember that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, which can help simplify complicated expressions!