2.jpeg "(8 – 5i)2")
Expanding (8 – 5i)<sup>2</sup>
This article will guide you through the process of expanding the expression (8 – 5i)<sup>2</sup>.
Understanding Complex Numbers
Before we dive into the expansion, let's recap what complex numbers are. A complex number is a number of the form a + bi, where:
- a and b are real numbers.
- i is the imaginary unit, defined as the square root of -1 (i<sup>2</sup> = -1).
Expanding the Expression
We can expand (8 – 5i)<sup>2</sup> using the following steps:
- Apply the square: (8 – 5i)<sup>2</sup> = (8 – 5i)(8 – 5i)
- Use the distributive property (FOIL method):
- First: 8 * 8 = 64
- Outer: 8 * (-5i) = -40i
- Inner: (-5i) * 8 = -40i
- Last: (-5i) * (-5i) = 25i<sup>2</sup>
- Combine like terms: 64 - 40i - 40i + 25i<sup>2</sup> = 64 - 80i + 25i<sup>2</sup>
- Substitute i<sup>2</sup> with -1: 64 - 80i + 25(-1) = 64 - 80i - 25
- Simplify: 39 - 80i
Final Result
Therefore, (8 – 5i)<sup>2</sup> expands to 39 - 80i.