The given tasks consist of arithmetic operations involving complex numbers. Below is the breakdown of the operations and general instructions:

Explanation of Tasks

1. Addition/Subtraction of Complex Numbers
   Form: $(a + bi) + (c + di)$ or $(a + bi) - (c + di)$,
   where real and imaginary parts are added/subtracted separately:
   $(a + c) + (b + d)i$

2. Multiplication of Complex Numbers
   Apply distributive property $(a + bi)(c + di)$:
   $(a + bi)(c + di) = ac + adi + bci + bdi^2$ Recall $i^2 = -1$.

3. Multiplication by a Real Number
   Multiply each part of the complex number by the real number.

4. Powers of Complex Numbers
   Use:

   • Square: Expand $(a + bi)^2$ using $(a + bi)(a + bi)$.
   • Cube: Expand step by step.

5. Expression with Differences of Squares
   Recognize $(a + b)(a - b) = a^2 - b^2$, where $i^2 = -1$.

Solutions of the Problems

I will solve a couple of tasks below and can proceed with others upon request.

1. (3 + 5i) + (7 - 2i)
   Add real and imaginary parts:
   $(3 + 7) + (5 - 2)i = 10 + 3i$

2. (-2 + 3i) - (7 - 2i)
   Subtract real and imaginary parts:
   $(-2 - 7) + (3 - (-2))i = -9 + 5i$

3. (3 - 2i) + (5 + i)
   Add real and imaginary parts:
   $(3 + 5) + (-2 + 1)i = 8 - i$

4. (-5 + 2i) + (5 + 2i)
   Add real and imaginary parts:
   $(-5 + 5) + (2 + 2)i = 0 + 4i \quad \text{or simply } 4i$

5. (2 + 3i)(5 - 7i)
   Use distributive property:
   $(2)(5) + (2)(-7i) + (3i)(5) + (3i)(-7i)$ Simplify each term:
   $10 - 14i + 15i - 21i^2$ Since $i^2 = -1$:
   $10 - 14i + 15i + 21 = 31 + i$

Would you like me to solve the rest or any specific part in detail? Let me know!

Related Questions

1. How do you multiply two complex numbers?
2. What is the process for squaring a complex number?
3. How does the imaginary unit $i$ work in powers like $i^2$ or $i^3$?
4. How can you identify and simplify expressions involving conjugates?
5. What is the modulus of a complex number?

Tip:

Remember that $i^2 = -1$ is a fundamental property in all complex number operations. It helps simplify terms involving $i^2$, $i^3$, etc.