https://file.aiquickdraw.com/mathbot/file/1729371114168skv6324f.jpg

The problem asks to simplify \((5 + 2i)(6 - 3i)\), where \(i\) represents the imaginary unit.

To solve this, we can apply the distributive property (also known as the FOIL method for binomials):

\[
(5 + 2i)(6 - 3i) = 5(6) + 5(-3i) + 2i(6) + 2i(-3i)
\]

Let's compute each term:

1. \( 5 \times 6 = 30 \)
2. \( 5 \times (-3i) = -15i \)
3. \( 2i \times 6 = 12i \)
4. \( 2i \times (-3i) = -6i^2 \)

Now, simplify:

- Combine the real terms and the imaginary terms: 
  \[
  30 + (-15i + 12i) + (-6i^2)
  \]
  - \(-15i + 12i = -3i\)
  - Remember that \(i^2 = -1\), so \(-6i^2 = -6(-1) = 6\).

Thus, the expression becomes:
\[
30 - 3i + 6 = 36 - 3i
\]

So the simplified result is \(36 - 3i\), which corresponds to the first option in the image.

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. Simplify \((4 + 5i)(3 - 2i)\).
2. What is the product of \((7 + i)(2 - 4i)\)?
3. How do you simplify \((5 - i)(6 + 2i)\)?
4. Multiply \((1 + 3i)(2 - 5i)\).
5. Simplify \((3 + 2i)(4 + 3i)\).

**Tip:** Remember that \(i^2 = -1\), which is essential when multiplying complex numbers!

Learn how to simplify the expression (5 + 2i)(6 - 3i) involving complex numbers using the FOIL method and key properties like i^2 = -1. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation