Let's go through the image and solve the mathematical problems presented:

Problem 7

The expression given for $A$ is:

$A = (-4)^2 \cdot 2^{-1} - 5 - (-3) \cdot 2^2 - (-2)^4$

1. First, solve each part step by step:

   • $(-4)^2 = 16$
   • $2^{-1} = \frac{1}{2} = 0.5$
   • $(-3) \cdot 2^2 = (-3) \cdot 4 = -12$
   • $(-2)^4 = 16$

2. Now substitute these values back: $A = 16 \cdot 0.5 - 5 - (-12) - 16$

3. Simplify: $A = 8 - 5 + 12 - 16 = -1$

Thus, $A = -1$.

Next, we have the expression for $B$:

$B = (-3)^3 - 2^2 \cdot 3 - 5^0 - 6^2 + (2+3) \cdot (6 - 4)^2$

1. Solve each part step by step:

   • $(-3)^3 = -27$
   • $2^2 = 4$, so $4 \cdot 3 = 12$
   • $5^0 = 1$
   • $6^2 = 36$
   • $(2+3) \cdot (6-4)^2 = 5 \cdot 2^2 = 5 \cdot 4 = 20$

2. Substitute these values: $B = -27 - 12 - 1 - 36 + 20$

3. Simplify: $B = -27 - 12 - 1 - 36 + 20 = -56$

Thus, $B = -56$.

Problem 8

The question asks to simplify the following expression:

$2^{15} + 2^{15}$

You can factor out $2^{15}$:

$2^{15} + 2^{15} = 2 \cdot 2^{15} = 2^{16}$

So the correct answer is $2^{16}$, which corresponds to option A.

Let me know if you need further details or clarifications!

5 Follow-Up Questions:

1. What is the rule for combining like terms with exponents, such as in $2^{15} + 2^{15}$?
2. How does the negative exponent, like $2^{-1}$, affect calculations?
3. Why is $5^0$ equal to 1, and what is the general rule for any number to the power of zero?
4. Can we simplify complex expressions with multiple variables using the same rules?
5. What are the key steps in simplifying an expression involving powers and multiplication?

Tip:

Always break down complex expressions into smaller, manageable steps, solving each part before putting everything together.