%5E2.jpeg "(5-3i)^2")
Expanding (5 - 3i)^2
In mathematics, especially complex numbers, squaring a complex number like (5 - 3i) involves expanding the expression and simplifying it. Here's how we do it:
Understanding Complex Numbers
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 (i² = -1).
Expanding the Expression
To square (5 - 3i), we multiply it by itself:
(5 - 3i)² = (5 - 3i)(5 - 3i)
Now, we use the distributive property (also known as FOIL method) to expand the expression:
(5 - 3i)(5 - 3i) = 5(5 - 3i) - 3i(5 - 3i)
= 25 - 15i - 15i + 9i²
Simplifying the Expression
We know that i² = -1. Substituting this into the equation:
25 - 15i - 15i + 9i² = 25 - 15i - 15i + 9(-1)
= 25 - 15i - 15i - 9
= 16 - 30i
The Result
Therefore, (5 - 3i)² simplifies to 16 - 30i.