(4+3zi)(6+5i)−(4+i)(6+5i)

Jasonbradley

2 min read

·

Jun 16, 2024

25

Share

Simplifying Complex Expressions

This article will explore the simplification of the complex expression: (4 + 3zi)(6 + 5i) - (4 + i)(6 + 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).

Simplifying the Expression

1. Expand the products:

   We begin by expanding the products using the distributive property (FOIL method):

   (4 + 3zi)(6 + 5i) = (4 * 6) + (4 * 5i) + (3zi * 6) + (3zi * 5i) = 24 + 20i + 18zi + 15zi²

   (4 + i)(6 + 5i) = (4 * 6) + (4 * 5i) + (i * 6) + (i * 5i) = 24 + 20i + 6i + 5i²

2. Simplify using i² = -1:

   We substitute i² with -1 in both expanded products:

   24 + 20i + 18zi + 15zi² = 24 + 20i + 18zi - 15z 24 + 20i + 6i + 5i² = 24 + 20i + 6i - 5

3. Combine like terms:

   Now we combine the real and imaginary terms separately:

   (24 - 15z) + (20 + 18z)i - (24 - 5) - (20 + 6)i

4. Simplify further:

   Simplifying the expression:

   (-15z + 5) + (18z - 14)i

Final Result

The simplified form of the expression (4 + 3zi)(6 + 5i) - (4 + i)(6 + 5i) is (-15z + 5) + (18z - 14)i.

This demonstrates a common method for simplifying complex expressions involving multiplication and subtraction. By applying the properties of complex numbers and algebraic manipulation, we can arrive at a simplified form that is easier to understand and work with.