(3---i)(1-%2B-2i)(1---i)(3-%2B-i).jpeg "(2 + I)(3 - I)(1 + 2i)(1 - I)(3 + I)")
Simplifying Complex Number Multiplication
This article will guide you through the simplification of the following complex number multiplication:
(2 + i)(3 - i)(1 + 2i)(1 - i)(3 + i)
We will utilize the distributive property and the fact that i² = -1 to simplify the expression.
Step 1: Multiply the first two factors
(2 + i)(3 - i) = 6 - 2i + 3i - i² = 6 + i + 1 = 7 + i
Step 2: Multiply the next two factors
(1 + 2i)(1 - i) = 1 - i + 2i - 2i² = 1 + i + 2 = 3 + i
Step 3: Multiply the result from Step 1 and Step 2
(7 + i)(3 + i) = 21 + 7i + 3i + i² = 21 + 10i - 1 = 20 + 10i
Step 4: Multiply the final factor
(20 + 10i)(3 + i) = 60 + 20i + 30i + 10i² = 60 + 50i - 10 = 50 + 50i
Final Result
Therefore, the simplified form of the given complex number multiplication is 50 + 50i.
This process demonstrates how to simplify complex number multiplications by utilizing the distributive property and the knowledge that i² = -1. By breaking down the problem into smaller steps, we can systematically simplify the expression to obtain the final result.