(2+3i)(3+5i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (2 + 3i)(3 + 5i)

This article will guide you through the process of multiplying two complex numbers: (2 + 3i) and (3 + 5i).

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 (i.e., i² = -1).

Multiplication Process

To multiply complex numbers, we use the distributive property (also known as FOIL method) just like we do with binomial expressions in algebra.

Here's how to multiply (2 + 3i)(3 + 5i):

1. Distribute:

   • (2 + 3i) * (3 + 5i) = 2(3 + 5i) + 3i(3 + 5i)

2. Simplify:

   • = (6 + 10i) + (9i + 15i²)

3. Substitute i² = -1:

   • = (6 + 10i) + (9i - 15)

4. Combine like terms:

   • = (6 - 15) + (10 + 9)i

5. Final result:

   • = -9 + 19i

Therefore, the product of (2 + 3i) and (3 + 5i) is -9 + 19i.

Summary

Multiplying complex numbers involves applying the distributive property and substituting i² with -1. This process leads to a new complex number, represented in the form a + bi.