(-5+5i)-(4-2i)+(-8-7i)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers

This article will guide you through the process of simplifying the complex expression: (-5 + 5i) - (4 - 2i) + (-8 - 7i)².

Understanding Complex Numbers

Complex numbers are numbers 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.

Simplifying the Expression

1. Expanding the Square:

   • Begin by squaring the complex number (-8 - 7i): (-8 - 7i)² = (-8 - 7i) * (-8 - 7i) = 64 + 56i + 56i + 49i² = 64 + 112i - 49 (since i² = -1) = 15 + 112i

2. Combining like terms:

   • Now, substitute the expanded square back into the original expression: (-5 + 5i) - (4 - 2i) + (15 + 112i)

   • Combine the real and imaginary terms separately: (-5 - 4 + 15) + (5 + 2 + 112)i

3. Final Simplification:

   • Simplify the expression: 6 + 129i

Conclusion

The simplified form of the complex expression (-5 + 5i) - (4 - 2i) + (-8 - 7i)² is 6 + 129i. This process involves understanding the properties of complex numbers, expanding squares, and combining like terms to arrive at a simplified form.