Let's break this expression down step by step:

You are working with:

\[
\left( \frac{16}{64} \right)^{\frac{1}{4}} \times \left( \frac{1}{4} \right)^{-2}
\]

### Step 1: Simplify \(\left( \frac{16}{64} \right)^{\frac{1}{4}}\)

\[
\frac{16}{64} = \frac{1}{4}
\]

So the expression becomes:

\[
\left( \frac{1}{4} \right)^{\frac{1}{4}} \times \left( \frac{1}{4} \right)^{-2}
\]

### Step 2: Apply the powers of fractions

For the first term:

\[
\left( \frac{1}{4} \right)^{\frac{1}{4}} = \left( \frac{1}{2^2} \right)^{\frac{1}{4}} = \frac{1}{2^{\frac{2}{4}}} = \frac{1}{2^{\frac{1}{2}}} = \frac{1}{\sqrt{2}}
\]

So now we have:

\[
\frac{1}{\sqrt{2}} \times \left( \frac{1}{4} \right)^{-2}
\]

### Step 3: Simplify \(\left( \frac{1}{4} \right)^{-2}\)

\[
\left( \frac{1}{4} \right)^{-2} = \frac{1}{(1/4)^2} = \frac{1}{1/16} = 16
\]

### Step 4: Multiply the results

Now we multiply:

\[
\frac{1}{\sqrt{2}} \times 16 = \frac{16}{\sqrt{2}}
\]

### Step 5: Rationalize the denominator

Multiply both numerator and denominator by \(\sqrt{2}\):

\[
\frac{16}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{16\sqrt{2}}{2} = 8\sqrt{2}
\]

### Final Answer:
\[
8\sqrt{2}
\]

Would you like more details on any of these steps or have any questions?

Here are 5 related questions you might find interesting:
1. How do you simplify expressions with negative exponents?
2. What are the steps to rationalizing a denominator?
3. How can fractional exponents be simplified?
4. How do you handle powers of fractions?
5. What is the importance of simplifying roots in mathematical expressions?

**Tip:** Always check if you can simplify fractions and expressions before applying more complex operations to avoid unnecessary calculations.

Learn how to simplify the expression (16/64)^1/4 x (1/4)^-2 with detailed steps. This problem involves working with exponents, simplifying fractions, and rationalizing denominators. Suitable for students in Grades 9-11, this guide covers exponent rules and fraction manipulation.

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