The image contains a mathematical expression:

$\frac{16^{\frac{3}{4}} \times \left( \frac{1}{8} \right)^{-1}}{4}$

We need to solve this expression step-by-step.

Step 1: Simplify $16^{\frac{3}{4}}$

$16 = 2^4$, so: $16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8$

Step 2: Simplify $\left( \frac{1}{8} \right)^{-1}$

The negative exponent means reciprocal, so: $\left( \frac{1}{8} \right)^{-1} = 8$

Step 3: Multiply the results

Now multiply the results from Step 1 and Step 2: $8 \times 8 = 64$

Step 4: Divide by 4

Finally: $\frac{64}{4} = 16$

Thus, the value of the expression is $\boxed{16}$.

Let me know if you would like further details or explanations.

Here are 5 related questions to expand your understanding:

1. How would the result change if the exponent on 16 was different, e.g. $16^{\frac{2}{3}}$?
2. What happens if you change the base of the fraction, say $\frac{1}{9}$, in the reciprocal step?
3. How does the property of negative exponents work in general?
4. Can you convert fractional exponents to radicals, and how would that affect the solution?
5. How would this problem change if the expression was divided by 8 instead of 4?

Tip: Always remember that negative exponents represent the reciprocal of the base raised to the positive of that exponent.