**Multiply the following complex numbers  Z1 = 5 + 3 ,   Z2 = 2i + 1**

To multiply the complex numbers \( Z_1 = 5 + 3i \) and \( Z_2 = 2i + 1 \), we use the distributive property (also known as the FOIL method for binomials):

\[
Z_1 \times Z_2 = (5 + 3i) \times (1 + 2i)
\]

Now, expand the expression:

\[
= 5 \cdot 1 + 5 \cdot 2i + 3i \cdot 1 + 3i \cdot 2i
\]

Simplify each term:

\[
= 5 + 10i + 3i + 6i^2
\]

Remember that \( i^2 = -1 \), so:

\[
= 5 + 13i + 6(-1)
\]

This simplifies to:

\[
= 5 + 13i - 6
\]

Combine the real parts:

\[
= -1 + 13i
\]

So, the product of the two complex numbers is:

\[
Z_1 \times Z_2 = -1 + 13i
\]

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

Here are 5 related questions:

1. How do you divide one complex number by another?
2. What is the magnitude of the complex number \( -1 + 13i \)?
3. How do you add or subtract complex numbers?
4. What does it mean to conjugate a complex number?
5. Can you multiply \( Z_1 \) and \( Z_2 \) in polar form?

**Tip:** Always remember that \( i^2 = -1 \) when multiplying complex numbers, as this will help you simplify the expression correctly.

Learn how to multiply complex numbers (5 + 3i) and (1 + 2i) using the distributive property. This detailed solution covers simplification steps and the application of the distributive property in complex number multiplication. Suitable for high school students studying complex numbers.

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