%5E2.jpeg "(5-4i)^2")
Squaring Complex Numbers: (5-4i)^2
This article explores the process of squaring the complex number (5-4i).
Understanding Complex Numbers
A complex number is a number 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.
Squaring (5-4i)
To square a complex number, we simply multiply it by itself:
(5-4i)^2 = (5-4i)(5-4i)
Now, we can expand this using the distributive property (FOIL method):
= 5(5-4i) - 4i(5-4i) = 25 - 20i - 20i + 16i^2
Since i^2 = -1, we can substitute:
= 25 - 20i - 20i - 16 = 9 - 40i
Conclusion
Therefore, (5-4i)^2 = 9 - 40i.
This result demonstrates the process of squaring complex numbers and how the imaginary unit 'i' interacts within these calculations.