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The expression provided is:

\[
(16x^{16}y^8)^{-\frac{1}{4}}
\]

Let's simplify this step-by-step, keeping only positive exponents:

### Step 1: Apply the exponent rule

When you raise a product to a power, you apply the power to each factor. The general rule is:

\[
(a \cdot b)^m = a^m \cdot b^m
\]

So:

\[
(16x^{16}y^8)^{-\frac{1}{4}} = 16^{-\frac{1}{4}} \cdot (x^{16})^{-\frac{1}{4}} \cdot (y^8)^{-\frac{1}{4}}
\]

### Step 2: Simplify each term

1. \( 16^{-\frac{1}{4}} \): The fourth root of 16 is 2, so:
   \[
   16^{-\frac{1}{4}} = \frac{1}{2}
   \]

2. \( (x^{16})^{-\frac{1}{4}} \): Use the power of a power rule, \( (a^m)^n = a^{m \cdot n} \), to multiply the exponents:
   \[
   (x^{16})^{-\frac{1}{4}} = x^{16 \cdot -\frac{1}{4}} = x^{-4}
   \]

3. \( (y^8)^{-\frac{1}{4}} \): Similarly, apply the power of a power rule:
   \[
   (y^8)^{-\frac{1}{4}} = y^{8 \cdot -\frac{1}{4}} = y^{-2}
   \]

### Step 3: Combine the results

Now combine the simplified terms:

\[
\frac{1}{2} \cdot x^{-4} \cdot y^{-2}
\]

### Step 4: Write with positive exponents

Since we want only positive exponents, we move \(x^{-4}\) and \(y^{-2}\) to the denominator:

\[
\frac{1}{2x^4y^2}
\]

Thus, the simplified expression is:

\[
\boxed{\frac{1}{2x^4y^2}}
\]

Would you like a detailed explanation of any step? 

Here are 5 questions to further explore this topic:
1. How would the simplification change if the exponent were \(\frac{1}{2}\) instead of \(-\frac{1}{4}\)?
2. What is the general rule for raising products with exponents to a power?
3. Can you simplify \( (25a^{10}b^6)^{\frac{1}{2}} \)?
4. What is the significance of using positive exponents in final answers?
5. How would you approach this problem if the exponent were positive?

**Tip:** Always apply the power of a power rule carefully by multiplying exponents together, and remember to convert negative exponents into positive ones by shifting terms across the fraction.

Simplify the following expression to simplest form using only positive exponents: (16x^16y^8)^(-1/4)

Simplify the algebraic expression (16x^16y^8)^(-1/4) using the power of a power rule and negative exponents. Follow a step-by-step solution to find the simplified form with only positive exponents, suitable for high school students (Grades 9-12).

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