(1-i)^3 In A+bi Form

Jasonbradley

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Jun 16, 2024

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Simplifying (1 - i)³ in the Form a + bi

This article will guide you through the process of simplifying the expression (1 - i)³ and expressing the result in the form a + bi, where a and b are real numbers.

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where:

• a is the real part of the complex number
• b is the imaginary part of the complex number
• i is the imaginary unit, defined as the square root of -1 (i² = -1)

Simplifying (1 - i)³

To simplify (1 - i)³, we can follow these steps:

1. Expand the cube: (1 - i)³ = (1 - i)(1 - i)(1 - i)

2. Multiply the first two factors: (1 - i)(1 - i) = 1 - i - i + i² = 1 - 2i - 1 = -2i

3. Multiply the result by the remaining factor: (-2i)(1 - i) = -2i + 2i² = -2i - 2 = -2 - 2i

Final Answer

Therefore, (1 - i)³ simplified to the form a + bi is -2 - 2i.