Question

Find one or more values of each of the following complex expressions in the easiest way you can. \((-i)^{1}\)

Short answer

-i

Step-by-step solution

- Understand the Expression

The given expression is \((-i)^1\). Note that any complex number raised to the power of 1 is itself.

- Simplify the Expression

Since any number raised to the first power is the number itself, therefore \((-i)^1 = -i\).

Key concepts

Imaginary Unit

The imaginary unit is a core concept in complex numbers. It is usually denoted as 'i' and has a unique property: \(i^2 = -1\). This unit is called 'imaginary' because it extends our number system beyond the real numbers.
When we are dealing with complex numbers, which have the form \(a + bi\) (where \(a\) and \(b\) are real numbers), the 'i' represents the imaginary part.
This enables us to solve equations that otherwise do not have solutions within the real number system, like \(x^2 = -1\).


**For example**:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

And since this pattern repeats, we can use it to simplify exponentiation involving 'i'. Remember, the value of any power of 'i' can be determined by its remainder when divided by 4.

Complex Exponentiation

Exponentiation in the context of complex numbers introduces an interesting twist to the familiar concept of raising numbers to powers. With complex numbers, we're often dealing with both real and imaginary parts.

For example, in the expression \((-i)^1\), since the exponent is 1, we simply revert to the base, \(-i\). The crucial point to remember with complex exponentiation is that the properties of the imaginary unit 'i' play a major role.
To illustrate:

• To compute \(i^4\), note that \(i^4 = (i^2)^2 = (-1)^2 = 1\).
• Similarly, \((-i)^1 = -i\) because any number raised to the power of 1 is itself.

This concept is highly useful when simplifying expressions involving higher powers of complex numbers.

Simplification

Simplification is key when working with complex numbers, as it helps to reduce complex expressions to their simplest forms for easier understanding and calculation. This involves understanding both the imaginary unit and complex exponentiation.
Let’s take the expression \(-i\)^1. Simplify it step-by-step:
First: Identify the base and exponent. Here, the base is \(-i\) and the exponent is 1.
Second: Apply basic exponent rules. Since any number raised to the power of 1 is the number itself, we get: \(-i\)^1 = -i.
By organizing the process into digestible steps, simplifying complex expressions becomes much more manageable, leading to clear solutions.


Remember these tips:

• Always start by identifying the exponent and the base.
• Use properties of the imaginary unit 'i' to simplify.
• Break down the problems into smaller, manageable parts.

Mastering these techniques will make working with complex numbers much simpler!