If 𝑧1 = 3 + 2𝑖 and 𝑧2 = 4 + 3𝑖. Find (i) 𝑧1𝑧2 (2 marks) (ii) 𝑧1 𝑧2 (2 marks)

Given the complex numbers \( z_1 = 3 + 2i \) and \( z_2 = 4 + 3i \), let's find:

### (i) \( z_1z_2 \) (2 marks)

To multiply two complex numbers, we use the distributive property:

\[
z_1z_2 = (3 + 2i)(4 + 3i)
\]

Now, distribute the terms:

\[
z_1z_2 = 3 \times 4 + 3 \times 3i + 2i \times 4 + 2i \times 3i
\]

Simplifying each term:

\[
z_1z_2 = 12 + 9i + 8i + 6i^2
\]

Since \( i^2 = -1 \):

\[
z_1z_2 = 12 + 17i + 6(-1)
\]

\[
z_1z_2 = 12 + 17i - 6
\]

\[
z_1z_2 = 6 + 17i
\]

### Answer for (i):

\[
z_1z_2 = 6 + 17i
\]

### (ii) \( \frac{z_1}{z_2} \) (2 marks)

To divide two complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator:

\[
\frac{z_1}{z_2} = \frac{3 + 2i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i}
\]

This simplifies to:

\[
\frac{z_1}{z_2} = \frac{(3 + 2i)(4 - 3i)}{(4 + 3i)(4 - 3i)}
\]

First, calculate the denominator using the difference of squares:

\[
(4 + 3i)(4 - 3i) = 4^2 - (3i)^2 = 16 - 9(-1) = 16 + 9 = 25
\]

Next, calculate the numerator:

\[
(3 + 2i)(4 - 3i) = 3 \times 4 + 3 \times (-3i) + 2i \times 4 + 2i \times (-3i)
\]

\[
= 12 - 9i + 8i - 6i^2
\]

Substitute \( i^2 = -1 \):

\[
= 12 - 9i + 8i + 6 = 18 - i
\]

Now, divide by the denominator:

\[
\frac{z_1}{z_2} = \frac{18 - i}{25} = \frac{18}{25} - \frac{1}{25}i
\]

### Answer for (ii):

\[
\frac{z_1}{z_2} = \frac{18}{25} - \frac{1}{25}i
\]

Would you like more details or have any questions?

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Here are five related questions:

1. How do you find the modulus and argument of a complex number like \( z_1 \) or \( z_2 \)?
2. What is the geometric interpretation of the multiplication of complex numbers?
3. How do you express the product \( z_1z_2 \) in polar form?
4. What is the conjugate of a complex number, and how is it used in division?
5. How do you add or subtract two complex numbers?

**Tip:** When dividing complex numbers, always multiply by the conjugate of the denominator to simplify the expression.

Learn how to solve complex number multiplication and division problems with detailed step-by-step solutions. This problem covers key concepts in complex numbers, including multiplication formulas and division using conjugates. Suitable for advanced high school students.

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