(1-i)^2(1+i)-(3-4i)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will guide you through simplifying the complex expression: (1-i)²(1+i) - (3-4i)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. Expand the squares:

   • (1-i)² = (1-i)(1-i) = 1 - 2i + i² = 1 - 2i - 1 = -2i
   • (3-4i)² = (3-4i)(3-4i) = 9 - 24i + 16i² = 9 - 24i - 16 = -7 - 24i

2. Substitute the expanded squares back into the original expression:

   • (-2i)(1+i) - (-7 - 24i)

3. Distribute and simplify:

   • -2i - 2i² + 7 + 24i = -2i + 2 + 7 + 24i = 9 + 22i

Final Result

Therefore, the simplified form of the expression (1-i)²(1+i) - (3-4i)² is 9 + 22i.