Let's work through each question from the image.

Question 1:

Problem: Simplify and put your answer in standard form with NO SPACES. $-i + (19 + 22i) - (8 - 14i)$

Solution:

1. Distribute the negative sign for the last term: $-i + (19 + 22i) - 8 + 14i$
2. Now, combine the real parts and imaginary parts separately:
   • Real parts: $19 - 8 = 11$
   • Imaginary parts: $-i + 22i + 14i = 35i$
3. Combine these results to get: $11 + 35i$

Answer: $11+35i$

Question 2:

Problem: Simplify $-9i^{65}$

Solution:

1. Recall that the powers of $i$ repeat in a cycle every four terms: $i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$
2. To find $i^{65}$, we divide 65 by 4 and look at the remainder: $65 \div 4 = 16 \text{ remainder } 1$ So, $i^{65} = i^1 = i$.
3. Now, substitute back: $-9i^{65} = -9i$

Answer: $-9i$

Do you want more details on these solutions or have any questions?

Here are some related questions for further practice:

1. How do you simplify expressions with complex numbers?
2. What is the pattern of powers of $i$ and why do they repeat every four terms?
3. How do you add and subtract complex numbers?
4. What does it mean for a complex number to be in standard form?
5. How do you multiply complex numbers and simplify the result?

Tip: Remember that the powers of $i$ cycle every four terms, so you can reduce high powers by finding the remainder when divided by 4.