The expression given in the image is:

$\frac{2^5 - 4 \cdot 3^3}{7 + (1 - \sqrt{100})}$

Let's simplify this step by step.

Step 1: Simplify the Numerator

First, compute the values inside the numerator.

• $2^5 = 32$
• $3^3 = 27$
• So, $4 \cdot 3^3 = 4 \cdot 27 = 108$

Now substitute these back into the numerator:

$2^5 - 4 \cdot 3^3 = 32 - 108 = -76$

Step 2: Simplify the Denominator

Next, let's simplify the denominator.

• $\sqrt{100} = 10$
• So, the denominator becomes $7 + (1 - 10)$

Simplify inside the parentheses:

$1 - 10 = -9$

Thus, the denominator is:

$7 + (-9) = 7 - 9 = -2$

Step 3: Combine the Numerator and Denominator

Now, put the simplified numerator and denominator together:

$\frac{-76}{-2} = 38$

Final Answer

The simplified expression is:

$\boxed{38}$

Would you like more details or have any questions?

Here are some additional related questions you might find helpful:

1. How do you calculate the value of higher powers such as $3^3$ and $2^5$?
2. What are some other examples of expressions that involve the square root?
3. Can you explain the process of simplifying expressions with multiple operations?
4. What rules apply when simplifying fractions?
5. How would the problem change if the denominator had been a positive number?

Tip: Always perform operations inside parentheses first and remember to simplify powers and roots before dealing with other operations.