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Let's work through the problem step by step.

### Given:
- \( z_1 = 3 + 4i \)
- \( z_2 = 10 + 5i \)

We need to solve the following:

---

### (a) Find the following results in the form \( x + yi \).

**(i) \( z_1 + z_2 \):**
\[
z_1 + z_2 = (3 + 4i) + (10 + 5i) = 13 + 9i
\]

**(ii) \( z_2 - z_1 \):**
\[
z_2 - z_1 = (10 + 5i) - (3 + 4i) = 7 + i
\]

**(iii) \( z_1 \cdot z_2 \):**
\[
z_1 \cdot z_2 = (3 + 4i)(10 + 5i)
\]
Use the distributive property:
\[
= 3(10 + 5i) + 4i(10 + 5i) = 30 + 15i + 40i - 20 = 10 + 55i
\]

**(iv) \( \frac{z_2}{z_1} \):**
\[
\frac{z_2}{z_1} = \frac{10 + 5i}{3 + 4i}
\]
Multiply numerator and denominator by the conjugate of the denominator:
\[
\frac{(10 + 5i)(3 - 4i)}{(3 + 4i)(3 - 4i)} = \frac{30 - 40i + 15i - 20}{9 + 16} = \frac{10 - 25i}{25}
\]
\[
= \frac{10}{25} - \frac{25i}{25} = 0.4 - i
\]

---

### (b) Find the following powers in the form \( x + yi \).

**(i) \( z_1^2 \):**
\[
z_1^2 = (3 + 4i)^2 = (3 + 4i)(3 + 4i)
\]
\[
= 9 + 12i + 12i - 16 = -7 + 24i
\]

**(ii) \( z_1^3 \):**
\[
z_1^3 = z_1^2 \cdot z_1 = (-7 + 24i)(3 + 4i)
\]
\[
= -21 - 28i + 72i + 96 = 75 + 44i
\]

---

### (c) Find the magn

Let z1 = 3 + 4i and z2 = 10 + 5i. Perform operations such as addition, subtraction, multiplication, division of complex numbers, powers, and magnitude calculations.

Learn how to perform arithmetic operations with complex numbers z1 = 3 + 4i and z2 = 10 + 5i. This includes addition, subtraction, multiplication, division, and powers of complex numbers. Understand key concepts like magnitudes and operations using conjugates.

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