%5E2.jpeg "(4-5i)^2")
Squaring a Complex Number: (4-5i)^2
In mathematics, 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.
Squaring a complex number means multiplying it by itself. Let's take a look at how to square the complex number (4-5i):
(4-5i)^2 = (4-5i)(4-5i)
To simplify this, we can use the distributive property (also known as FOIL):
(4-5i)(4-5i) = 4(4-5i) - 5i(4-5i)
Expanding this gives us:
16 - 20i - 20i + 25i^2
Since i^2 = -1, we can substitute:
16 - 20i - 20i + 25(-1)
Combining the real and imaginary terms:
(16 - 25) + (-20 - 20)i
Finally, we get:
**(4-5i)^2 = ** -9 - 40i
Therefore, the square of the complex number (4-5i) is -9 - 40i.
This process can be applied to square any complex number, just remember to use the distributive property and substitute i^2 with -1.