The expression given in the image is:

$-7^2 - (-2)^3$

Step-by-Step Solution:

1. Evaluate the exponentiation:

   • $7^2 = 7 \times 7 = 49$
   • $(-2)^3 = (-2) \times (-2) \times (-2) = -8$

2. Apply the signs:

   • The first term is $-7^2$, which is $-49$ (because the negative sign in front is not affected by the exponent).
   • The second term is $-(-2)^3$, which is $-(-8) = 8$.

3. Combine the results:

   $-49 + 8 = -41$

Final Answer:

$-41$

Rules Used:

1. Order of Operations (PEMDAS/BODMAS): Exponentiation is performed before multiplication, division, addition, and subtraction.
2. Handling Negative Signs: The negative sign in front of a number is applied after the exponentiation. For example, $-7^2$ means you square 7 first (getting 49), then apply the negative sign to get $-49$.

Would you like more details on any step or have any questions?

Related Questions:

1. What is the result of $-5^2 - 3^3$?
2. How does the order of operations affect expressions with multiple exponents?
3. What is the value of $(-4)^2 - (-3)^3$?
4. How would the result differ if the expression was $(-7)^2 - (-2)^3$?
5. Can you solve $-6^2 + 2^4$ and explain the steps?

Tip:

Always apply exponents before applying the negative sign if the negative sign is outside the parentheses.