%5E2-(4-5i)%5E2.jpeg "(4+5i)^2-(4-5i)^2")
Expanding and Simplifying (4+5i)^2 - (4-5i)^2
This article explores the simplification of the expression (4+5i)^2 - (4-5i)^2, demonstrating the use of complex number arithmetic.
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, where i^2 = -1.
Expanding the Expression
Let's start by expanding each term individually:
- (4+5i)^2: Using the FOIL method (First, Outer, Inner, Last), we get: (4+5i)(4+5i) = 16 + 20i + 20i + 25i^2 = 16 + 40i - 25 = -9 + 40i
- (4-5i)^2: Again, using FOIL: (4-5i)(4-5i) = 16 - 20i - 20i + 25i^2 = 16 - 40i - 25 = -9 - 40i
Final Simplification
Now, we can substitute the expanded forms back into the original expression:
(4+5i)^2 - (4-5i)^2 = (-9 + 40i) - (-9 - 40i)
Simplifying further:
= -9 + 40i + 9 + 40i = 80i
Conclusion
Therefore, the simplified form of (4+5i)^2 - (4-5i)^2 is 80i. This example highlights the importance of understanding complex number arithmetic, particularly the expansion and simplification of expressions involving complex numbers.