(1-i)^3 In Standard Form

Jasonbradley

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Jun 16, 2024

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Simplifying (1 - i)³ in Standard Form

This article will guide you through the process of simplifying the complex number (1 - i)³ and expressing it in standard form (a + bi).

Understanding Complex Numbers

A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).

Simplifying (1 - i)³

To simplify (1 - i)³, we can use the following steps:

1. Expand using the binomial theorem:

   (1 - i)³ = (1 - i)(1 - i)(1 - i)

   Expanding the first two terms:

   = (1 - 2i + i²) (1 - i)

2. Substitute i² = -1:

   = (-2i) (1 - i)

3. Expand and simplify:

   = -2i + 2i² = -2i - 2

4. Express in standard form (a + bi):

   = -2 - 2i

Final Answer

Therefore, (1 - i)³ simplified in standard form is -2 - 2i.