Problem 9 Calculate the given expression. ... [FREE SOLUTION]

Chapter 6: Problem 9

Calculate the given expression. \(i^{3}\)

Short Answer

Expert verified

The answer is \(-i\).

Step by step solution

Identify the Given Expression

The expression to calculate is \(i^3\). Here, \(i\) is the imaginary unit, which is equal to the square root of -1, i.e., \(i = \sqrt{-1}\).

Recall the Powers of \(i\)

Recall the key powers of \(i\): \[ i^1 = i \] \[ i^2 = -1 \] \[ i^3 = i \cdot i^2 = i \cdot (-1) \] \[ i^4 = 1 \] These powers repeat every four exponents.

Calculate \(i^3\)

From the key powers of \(i\), substitute \(i^2\) in the expression for \(i^3\): \[ i^3 = i \cdot i^2 = i \cdot (-1) = -i \] Therefore, \(i^3\) is \(-i\).

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Key Concepts

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

Imaginary Unit

In mathematics, the imaginary unit is a number denoted by the symbol 'i'. It is defined by the property that its square equals -1. This means that:
\[ i^2 = -1 \]
The imaginary unit is a fundamental component in the field of complex numbers, which extend the real number system to include solutions to equations like x虏 + 1 = 0, which have no solutions within the set of real numbers. The concept of 'i' helps in exploring phenomena such as oscillations and waves where real numbers fall short.
Using the imaginary unit, we can express more complex numbers in a simple form.

Powers of i

Understanding the powers of the imaginary unit 'i' is crucial for dealing with complex numbers. The powers of 'i' follow a cyclical pattern:

\[ i^1 = i \]
\[ i^2 = -1 \]
\[ i^3 = i \times i^2 = i \times (-1) = -i \]
\[ i^4 = i \times i^3 = i \times (-i) = -i^2 = -(-1) = 1 \]
After \ i^4 = 1\, the cycle repeats:
\[ i^5 = i \times i^4 = i \times 1 = i \]
\[ i^6 = i \times i^5 = i \times i = -1 \]
Recognizing this periodic pattern makes it easier to calculate higher powers of 'i' without doing extensive multiplication each time.

Complex Numbers

Complex numbers are an extension of the real numbers, introduced to solve equations that have no real solutions. A complex number has the form:
\[ a + bi \] where:

\ a\ is the real part.
\ bi\ is the imaginary part, with \ b\ being a real number and \ i\ the imaginary unit.

For example, \ 3 + 4i \ is a complex number with real part 3 and imaginary part 4i.
Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules, where you apply the distributive and associative properties. Importantly, when performing these operations, remember that \[ i^2 = -1. \]
Complex numbers are vital in various fields of engineering, physics, and applied mathematics, particularly in signal processing and fluid dynamics.

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91影视

Calculate the given expression. \(i^{3}\)

Short Answer

Step by step solution

Identify the Given Expression

Recall the Powers of \(i\)

Calculate \(i^3\)

Key Concepts

Imaginary Unit

Powers of i

Complex Numbers

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