The expression from the image is:

$\frac{36 \frac{1}{2}}{27 \frac{2}{3} - \left( \frac{1}{2} \right)^{-2}}$

Let's break this down step by step:

Step 1: Convert mixed fractions to improper fractions.

• $36 \frac{1}{2} = \frac{73}{2}$
• $27 \frac{2}{3} = \frac{83}{3}$

Step 2: Simplify the exponent.

The expression $\left( \frac{1}{2} \right)^{-2}$ means we take the reciprocal of $\frac{1}{2}$ and square it:

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

Step 3: Substitute and simplify.

Now, the expression becomes:

$\frac{\frac{73}{2}}{\frac{83}{3} - 4}$

Step 4: Combine the terms in the denominator.

To subtract $4$ from $\frac{83}{3}$, we write $4$ as $\frac{12}{3}$:

$\frac{83}{3} - 4 = \frac{83}{3} - \frac{12}{3} = \frac{71}{3}$

Step 5: Simplify the overall fraction.

The expression now looks like:

$\frac{\frac{73}{2}}{\frac{71}{3}} = \frac{73}{2} \times \frac{3}{71} = \frac{73 \times 3}{2 \times 71} = \frac{219}{142}$

Thus, the simplified result is:

$\frac{219}{142}$

Would you like any additional details on this problem? Here are some follow-up questions you might find helpful:

1. How do you convert improper fractions to mixed numbers?
2. What is the method for simplifying complex fractions?
3. How do negative exponents work for fractions?
4. Can you apply this to solving similar expressions?
5. What are some tips for handling subtraction in fractional form?

Tip: When simplifying fractions involving exponents, always resolve the exponent first to avoid confusion in the order of operations.