The expression in the image is:

$\left( -\frac{1}{2} \right)^{-2} \times 2^2 \div \left( \frac{1}{2} \right)^{-5} + \left( \sqrt{7} \right)^0$

Let's break this down step by step:

1. Evaluate $\left( -\frac{1}{2} \right)^{-2}$: Using the rule $(a^{-n}) = \frac{1}{a^n}$, we get: $\left( -\frac{1}{2} \right)^{-2} = \frac{1}{\left( -\frac{1}{2} \right)^2} = \frac{1}{\frac{1}{4}} = 4$

2. Evaluate $2^2$: $2^2 = 4$

3. Evaluate $\left( \frac{1}{2} \right)^{-5}$: Again using the same rule: $\left( \frac{1}{2} \right)^{-5} = \frac{1}{\left( \frac{1}{2} \right)^5} = \frac{1}{\frac{1}{32}} = 32$

4. Evaluate $\left( \sqrt{7} \right)^0$: Any number raised to the power of 0 is 1: $\left( \sqrt{7} \right)^0 = 1$

Now substitute these values into the expression:

$4 \times 4 \div 32 + 1$

5. Simplify:

First, calculate $4 \times 4 = 16$.

Next, divide $16 \div 32 = \frac{1}{2}$.

Now add $1$:

$\frac{1}{2} + 1 = \frac{3}{2}$

So, the final result is:

$\boxed{\frac{3}{2}}$

Would you like more details on any step? Here are some related questions:

1. What are the laws of exponents used in solving this?
2. How do you handle negative exponents in different scenarios?
3. Can you simplify a complex expression involving radicals and powers?
4. What is the rule for raising any number to the power of zero?
5. How do you perform division involving fractions and powers?

Tip: Always simplify exponents before doing multiplication or division to make calculations easier.