(6-8i)^2

2 min read Jun 16, 2024
(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.

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