Problem 14 - Contemporary Precalculus

Question

In Exercises \(9-14,\) find the absolute value. $$\left|i^{7}\right|$$

Step-by-Step Solution

Answer

Answer: 1

Step 1: Simplify i^7

Before finding the absolute value, we need to simplify i^7. Remember that i is an imaginary unit, which is defined as the square root of -1. Therefore, we have: \(i^2 = -1\) Let's simplify i^7 using the properties of exponents: \(i^7 = i^{(4)(1)+3} = (i^4)^1 * i^3\) Since \( i^4 = (i^2)^2 = 1\), we get: \(i^7 = 1 * i^3 = i^3\) Now, we need to find the absolute value of i^3.

Step 2: Calculate the absolute value of i^3

The absolute value of a complex number is the distance from the origin (0,0) to the point representing the complex number in the complex plane. In our case, i^3 can be expressed in terms of i^2 as follows: \(i^3 = i^2 * i = (-1) * i = -i\) Now, let's find the absolute value of -i: \(\left|-i\right| = \sqrt{(-0)^2 + (-1)^2} = \sqrt{0 + 1} = \sqrt{1} = 1\) So, the absolute value of i^7 is 1.

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers used to solve equations that do not have solutions in the realm of real numbers alone, such as the equation \(x^2 + 1 = 0\). A complex number is typically expressed as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. Complex numbers combine both a real component, \(a\), and an imaginary component, \(bi\).

To understand how complex numbers behave, it's important to remember they can be represented on a two-dimensional plane known as the complex plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. This helps in visualizing operations like addition and multiplication for complex numbers.

• **Addition** involves adding the real parts together and adding the imaginary parts together.
• **Multiplication** is a bit trickier, requiring distribution and using the fact that \(i^2 = -1\).

These operations are fundamental when working with complex numbers, as they allow you to manipulate them just like any other numbers, albeit with these additional properties.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a mathematical concept created to extend the set of real numbers. It is defined as the square root of \(-1\).

In equations, you will often see \(i\) used to represent this solution to the equation \(x^2 = -1\). Although it might seem abstract at first, the imaginary unit is incredibly useful for solving mathematical problems that have no real solutions.

• The power of \(i\) follows a cyclical pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats.
• This cyclical nature is crucial for simplifying expressions involving \(i\) raised to higher powers, as seen in the solution to \(\left|i^7\right|\), simplified as \(\left|i^3\right|\).

The imaginary unit, therefore, is more than a theoretical construct—it's a practical tool for dozens of mathematical operations, especially those involving complex numbers.

Properties of Exponents

Understanding the properties of exponents is essential for simplifying mathematical expressions and solving equations. These properties provide a systematic way to manipulate powers of numbers, including the imaginary unit \(i\).

There are several key rules of exponents that are useful across various applications:

• **Product of Powers Rule**: \(a^m \cdot a^n = a^{m+n}\)
• **Power of a Power Rule**: \((a^m)^n = a^{m \cdot n}\)
• **Power of a Product Rule**: \((ab)^n = a^n \cdot b^n\)

These properties help in systematically breaking down complex expressions into more manageable forms. In the context of the original exercise, it was particularly useful to understand how \(i^7\) can be broken down into \((i^4)^1 \times i^3\) because \(i^4 = 1\), making the simplification process easy.

Mastering the properties of exponents allows you to simplify expressions efficiently and is applicable in various fields such as algebra, calculus, and even physics, where the manipulation of exponential terms is a frequent requirement.