-2(1-3i))%5E2.jpeg "((5-i)-2(1-3i))^2")
Simplifying the Complex Expression: ((5-i)-2(1-3i))^2
This article will guide you through the steps to simplify the complex expression ((5-i)-2(1-3i))^2. Let's break it down step by step:
1. Distribute and Simplify Inside the Parentheses
First, we need to distribute the -2 inside the second set of parentheses:
(5 - i) - 2(1 - 3i) = (5 - i) - 2 + 6i
Then, combine the real and imaginary terms:
(5 - i) - 2 + 6i = (5 - 2) + (-1 + 6)i = 3 + 5i
Now, our expression simplifies to: (3 + 5i)^2
2. Expand the Square
To expand (3 + 5i)^2, we use the FOIL method (First, Outer, Inner, Last):
(3 + 5i)^2 = (3 + 5i)(3 + 5i)
= (3 * 3) + (3 * 5i) + (5i * 3) + (5i * 5i)
= 9 + 15i + 15i + 25i^2
3. Substitute and Simplify
Remember that i^2 = -1. Substitute this into our expression:
9 + 15i + 15i + 25i^2 = 9 + 15i + 15i + 25(-1)
= 9 + 15i + 15i - 25
Finally, combine the real and imaginary terms:
= (9 - 25) + (15 + 15)i
4. The Simplified Result
The simplified form of ((5-i)-2(1-3i))^2 is:
-16 + 30i