(3-2i^3)-(8+i^2)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (3 - 2i^3) - (8 + i^2)

This article will guide you through the process of simplifying the complex expression: (3 - 2i^3) - (8 + i^2).

Understanding the Basics

Before we delve into the simplification, let's refresh our understanding of complex numbers:

• Complex Number: 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.
• Imaginary Unit (i): The imaginary unit 'i' is defined as the square root of -1 (i.e., i² = -1).

Simplifying the Expression

Now, let's break down the simplification step-by-step:

1. Simplify the powers of i:

   • i^3 = i^2 * i = -1 * i = -i
   • i^2 = -1

2. Substitute the simplified powers of i: (3 - 2i^3) - (8 + i^2) becomes (3 - 2(-i)) - (8 + (-1))

3. Distribute the negative sign: (3 + 2i) - (8 - 1)

4. Combine the real and imaginary terms: (3 - 8 + 1) + (2i)

5. Final Simplified Expression: -4 + 2i

Conclusion

Therefore, the simplified form of the complex expression (3 - 2i^3) - (8 + i^2) is -4 + 2i. This illustrates how understanding the properties of complex numbers, specifically the imaginary unit 'i', allows us to simplify expressions effectively.