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Chapter 2: Problem 84

Simplifying a Complex Number. Simplify the complex number and write it in standard form. $$(-i)^{3}$$

Short Answer

Expert verified
The simplified form of the complex number \((-i)^3\) is \( i \).

Step by step solution

01

Understand the Behavior of \( i \) When Raised to Various Powers

The complex number \( i \) is defined as the square root of -1. This means that \( i^2 = -1 \). Now, when you cube \( i \), i.e. \( i^3 \), you are effectively multiplying \( i^2 \) by \( i \). 'i' to the power of 1, 2, 3, and 4 have the values \( i, -1, -i \), and 1 respectively. This is a cycle which repeats for every four powers of 'i'.
02

Apply the Behavior of \( i \) to Your Given Power

Since the base of the power is -i, you can consider the cube of -i, which is \((-i)^3 = -i^3 \). Now, if you look from the defined cycle under step 1, \( i^3 = -i \). So, by substitution, \((-i)^3 = -(-i) \).
03

Simplify the Complex Number

Now, simplifying the double negative will result in 'i'. This is the standard form of the complex number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Number Standard Form
When working with complex numbers, it's essential to know their standard form, which is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The coefficients \(a\) and \(b\) are real numbers, and \(i\) represents the square root of -1.

For instance, if you have the expression \(3 - 4i\), that's already in standard form with \(a = 3\) and \(b = -4\). Simplifying complex numbers often means performing operations to bring them into this clear, standardized format.
Properties of i
The imaginary unit \(i\) has unique properties that are fundamental to working with complex numbers. The most critical property is that \(i^2 = -1\). This characteristic leads to a pattern when \(i\) is raised to higher powers:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = i^2 \times i = -1 \times i = -i\)
  • \(i^4 = (i^2)^2 = (-1)^2 = 1\)

This cyclical nature continues endlessly, so for any integer power \(n\), we can simplify \(i^n\) by finding its remainder when divided by 4 and substituting the respective value from the cycle. This property is key when simplifying expressions involving powers of \(i\).
Raising i to Various Powers
When raising \(i\) to any power, it鈥檚 helpful to remember the cycle of four. The powers of \(i\) are periodic with a period of 4. That means after every fourth power, the values start repeating. Here鈥檚 what the cycle looks like:
  • \(i^0 = 1\) (Any number to the power of 0 is 1)
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\) (And now the cycle repeats)

Understanding this cycle allows us to simplify any power of \(i\) by finding the remainder of the exponent divided by 4. For example, \(i^{3}\) will loop back through the cycle once to \(i\), and two more steps bring us to \(i^3 = -i\).
Imaginary Numbers
Imaginary numbers are the numbers that give a negative product when squared. They're written as a real number multiplied by the imaginary unit \(i\). The concept of imaginary numbers was introduced to solve equations that do not have solutions in the set of real numbers, such as \(x^2 + 1 = 0\), where \(x\) would be \(i\), the principal square root of -1.

Imaginary numbers, when combined with real numbers, form complex numbers, expanding the field of possible solutions in mathematics. It's why an expression like \(2i + 3\) is a complex number, with its real part being 3 and its imaginary part being 2i.

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Most popular questions from this chapter

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Population Growth The game commission introduces 100 deer into newly acquired state game lands. The population \(N\) of the herd is modeled by $$N=\frac{20(5+3 t)}{1+0.04 t}, \quad t \geq 0$$ where \(t\) is the time in years. (a) Use a graphing utility to graph the model. (b) Find the populations when \(t=5, t=10\) , and \(t=25 .\) (c) What is the limiting size of the herd as time increases?
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