(2+4i^5)+(1-9i^6)-(3+i^7)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (2 + 4i^5) + (1 - 9i^6) - (3 + i^7)

This article explores the simplification of the complex expression (2 + 4i^5) + (1 - 9i^6) - (3 + i^7). We will break down the steps involved and understand the properties of imaginary numbers.

Understanding Imaginary Numbers

The imaginary unit i is defined as the square root of -1 (i.e., i² = -1). This definition allows us to work with the square roots of negative numbers. Higher powers of i follow a cyclical pattern:

• i³ = i² * i = -1 * i = -i
• i⁴ = (i²)² = (-1)² = 1
• i⁵ = i⁴ * i = 1 * i = i
• i⁶ = i⁴ * i² = 1 * -1 = -1

And so on. This pattern repeats every four powers.

Simplifying the Expression

1. Identify and Simplify Powers of i:

   • i^5 = i⁴ * i = 1 * i = i
   • i^6 = i⁴ * i² = 1 * -1 = -1
   • i^7 = i⁶ * i = -1 * i = -i

2. Substitute the simplified powers of i into the expression: (2 + 4i) + (1 - 9(-1)) - (3 - i)

3. Simplify the expression by combining real and imaginary terms: (2 + 1 + 9 - 3) + (4 + 1)i

4. Combine like terms: 9 + 5i

Conclusion

The simplified form of the complex expression (2 + 4i^5) + (1 - 9i^6) - (3 + i^7) is 9 + 5i. This process demonstrates how understanding the properties of imaginary numbers and their cyclic patterns allows us to simplify complex expressions effectively.