https://file.aiquickdraw.com/mathbot/file/17264175862228v1vi2kr.jpg

The problem asks for the modulus of the complex number \( z = (3 - 4i)(12 + 5i) \).

### Steps:

1. **Multiply the two complex numbers**:
   We need to use the distributive property (FOIL method):
   \[
   z = (3 - 4i)(12 + 5i)
   \]

   Expanding:
   \[
   z = 3(12) + 3(5i) - 4i(12) - 4i(5i)
   \]
   \[
   z = 36 + 15i - 48i - 20i^2
   \]
   
   Since \( i^2 = -1 \), replace \( -20i^2 \) with \( 20 \):
   \[
   z = 36 + 15i - 48i + 20
   \]
   Combine like terms:
   \[
   z = (36 + 20) + (15i - 48i)
   \]
   \[
   z = 56 - 33i
   \]

2. **Find the modulus**:
   The modulus of a complex number \( z = a + bi \) is given by:
   \[
   |z| = \sqrt{a^2 + b^2}
   \]
   For \( z = 56 - 33i \), we have \( a = 56 \) and \( b = -33 \).

   Now calculate:
   \[
   |z| = \sqrt{56^2 + (-33)^2}
   \]
   \[
   |z| = \sqrt{3136 + 1089}
   \]
   \[
   |z| = \sqrt{4225}
   \]
   \[
   |z| = 65
   \]

### Final Answer:
The modulus of the complex number \( z = (3 - 4i)(12 + 5i) \) is \( 65 \).

Would you like further details or have any questions?

Here are some related questions you might consider:
1. How do you find the modulus of a general complex number?
2. Can you explain the geometric interpretation of the modulus of a complex number?
3. What are the rules for multiplying two complex numbers?
4. How does the imaginary unit \( i \) behave in powers greater than 2?
5. How can you use complex numbers in solving quadratic equations?

**Tip:** Always check if you’ve combined like terms correctly when simplifying complex numbers, especially when multiplying.

Calculați modulul numărului complex z = (3 - 4i)(12 + 5i).

Learn how to calculate the modulus of the complex number z = (3 - 4i)(12 + 5i) using the distributive property and modulus formula. Suitable for students in Grades 10-12, this solution covers key concepts in complex number multiplication and modulus calculation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation