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(8-2i^4)+(3-7i^8)-(4+i^9)

2 min read Jun 16, 2024
(8-2i^4)+(3-7i^8)-(4+i^9)

Simplifying Complex Expressions: (8-2i^4)+(3-7i^8)-(4+i^9)

This article will walk through the process of simplifying the complex expression: (8-2i^4)+(3-7i^8)-(4+i^9).

Understanding the Properties of Imaginary Numbers

Before we can simplify the expression, we need to recall some key properties of imaginary numbers:

  • i is defined as the square root of -1.
  • i^2 = -1
  • i^3 = i^2 * i = -1 * i = -i
  • i^4 = i^2 * i^2 = (-1) * (-1) = 1
  • This pattern of i^n repeats every four powers.

Simplifying the Expression

  1. Simplify the powers of i:

    • i^4 = 1
    • i^8 = (i^4)^2 = 1^2 = 1
    • i^9 = i^4 * i^5 = 1 * i * (i^4) = i

  2. Substitute the simplified values:

    • (8 - 2 * 1) + (3 - 7 * 1) - (4 + i)

  3. Simplify the real and imaginary components:

    • (8 - 2 + 3 - 7 - 4) + (-1)i

  4. Combine the terms:

    • 0 - i

Conclusion

Therefore, the simplified form of the complex expression (8-2i^4)+(3-7i^8)-(4+i^9) is -i.

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