(-i)(2i)(-1/8i)^3

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: A Step-by-Step Guide

This article will guide you through the process of simplifying the complex expression (-i)(2i)(-1/8i)^3. We will break down each step, explaining the rules of complex numbers and how they apply to this particular example.

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.

Here are some key properties of complex numbers we'll use:

• i² = -1
• (a + bi)(c + di) = (ac - bd) + (ad + bc)i (distributive property)

Simplifying the Expression

Let's break down the expression (-i)(2i)(-1/8i)^3 step-by-step:

1. Simplify the cube:

   • (-1/8i)^3 = (-1/8i) * (-1/8i) * (-1/8i) = -1/512i³

2. Apply i² = -1:

   • -1/512i³ = -1/512 * i² * i = -1/512 * (-1) * i = 1/512i

3. Multiply the remaining terms:

   • (-i)(2i)(1/512i) = -2/512 i³ = -2/512 * (-1) * i = 1/256i

Therefore, the simplified form of the expression (-i)(2i)(-1/8i)^3 is 1/256i.

Conclusion

Simplifying complex expressions can seem daunting, but by applying the rules of complex numbers and breaking down the expression step by step, it becomes a manageable task. Remember to utilize the properties of complex numbers like i² = -1 and the distributive property to arrive at the simplified form.