%5E2.jpeg "(6-8i)^2")
Squaring Complex Numbers: A Step-by-Step Guide for (6 - 8i)²
This article will explore how to square the complex number (6 - 8i). We'll break down the process into clear, manageable steps, and explain the key concepts involved.
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 (i² = -1).
Squaring a Complex Number
To square a complex number, we simply multiply it by itself.
Let's apply this to our example:
(6 - 8i)² = (6 - 8i) * (6 - 8i)
Now, we'll expand this product using the FOIL (First, Outer, Inner, Last) method:
- First: 6 * 6 = 36
- Outer: 6 * (-8i) = -48i
- Inner: (-8i) * 6 = -48i
- Last: (-8i) * (-8i) = 64i²
Combining the terms, we get:
36 - 48i - 48i + 64i²
Remember that i² = -1. Substitute this into our expression:
36 - 48i - 48i + 64(-1)
Simplify by combining like terms:
36 - 96i - 64
Finally, combine the real and imaginary components:
(36 - 64) + (-96)i
This simplifies to:
-28 - 96i
Conclusion
Therefore, (6 - 8i)² equals -28 - 96i. This process demonstrates how to square a complex number by applying the fundamental rules of complex arithmetic. Understanding these steps is crucial for working with complex numbers in various mathematical fields.