(3+3i)(1+4i)(-2+3i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers

This article will explore the multiplication of three complex numbers: (3 + 3i)(1 + 4i)(-2 + 3i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1.

Multiplication Process

To multiply complex numbers, we can use the distributive property. Let's break down the multiplication step-by-step:

1. Multiply the first two complex numbers: (3 + 3i)(1 + 4i) = 3(1 + 4i) + 3i(1 + 4i) = 3 + 12i + 3i + 12i² = 3 + 15i + 12i²

2. Substitute i² with -1: = 3 + 15i + 12(-1) = -9 + 15i

3. Multiply the result with the third complex number: (-9 + 15i)(-2 + 3i) = -9(-2 + 3i) + 15i(-2 + 3i) = 18 - 27i - 30i + 45i² = 18 - 57i + 45i²

4. Substitute i² with -1 again: = 18 - 57i + 45(-1) = -27 - 57i

Final Result

Therefore, the product of (3 + 3i)(1 + 4i)(-2 + 3i) is -27 - 57i.

Key Takeaways

• Complex number multiplication involves using the distributive property and replacing i² with -1.
• Complex numbers can be represented in the form a + bi, where 'a' and 'b' are real numbers.
• The multiplication of complex numbers results in another complex number.