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.