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Chapter 8: Problem 470

Simplify. \(i^{39}\)

Short Answer

Expert verified
The simplified form of \(i^{39}\) is \(-i\).

Step by step solution

01

Understand the Properties of i

Recall that the imaginary unit, denoted by i, is defined by the property that \(i^2 = -1\). Additionally, it follows that \(i^3 = i \times i^2 = i \times (-1) = -i\) and \(i^4 = (i^2)^2 = (-1)^2 = 1\). Notice that these powers of i cycle every 4 steps: \(i, -1, -i, 1\).
02

Determine the Remainder When Dividing by 4

Given any power of i, find the exponent modulo 4 to determine its equivalent simplified form. Calculate \(39 \mod 4\). When dividing 39 by 4, the quotient is 9 with a remainder of 3, since \39 = 4 \times 9 + 3\. Therefore, \39 \mod 4 = 3\.
03

Use the Remainder to Simplify the Expression

From the previous step, since \39 \mod 4 = 3\, we align this with the known cycle of powers of i: \(i, -1, -i, 1\). Hence, \(i^{39} = i^3 = -i\).

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Key Concepts

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

Powers of i
The imaginary unit, denoted as \(i\), plays a fundamental role in complex numbers. By definition, \(i\) is the square root of -1. This means \(i^2 = -1\). This special property leads to a cyclical pattern in the powers of \(i\), which repeats every four steps.
Here is the pattern:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
After these four values, the cycle repeats. For example, \(i^5 = i\), \(i^6 = -1\), and so on. Recognizing this cycle is key to simplifying higher powers of \(i\). Just identify the position within the cycle and immediately find the simplified value.
Modulo Operation
To simplify expressions involving high powers of \(i\), we use the modulo operation. Modulo operation finds the remainder after division of one number by another. For our purposes, we use modulo 4 because the powers of \(i\) cycle every four steps.
Here's how it works:
Consider the expression \(i^{39}\). To simplify \(i^{39}\), we determine the remainder when 39 is divided by 4.
Calculate:
\[39 \, \text{mod} \, 4 = 3\]
This means \(i^{39}\) corresponds to the same position in the cycle as \(i^3\).
Therefore, \(i^{39}\) simplifies to \(-i\).
Using the modulo operation can turn potentially complex calculations into very manageable ones.
Simplifying Expressions
Simplifying expressions involving imaginary numbers often involves recognizing patterns and applying straightforward operations. Let's walk through the simplification of \(i^{39}\) step-by-step.
  • **Identify the cycle of \(i\):**
    We know the cycle is \(i, -1, -i, 1\).
  • **Use modulo operation to find the position in the cycle:**
    Calculate \[39 \, \text{mod} \, 4 = 3\]
  • **Match the remainder with the power of \(i\):**
    Since \(3\) corresponds to \(i^3\) in the cycle, we identify \(i^{39}\) as \(i^3\).
  • **Simplify the power using the cycle:**
    Recall from the cycle that \(i^3 = -i\).
So, \(i^{39}\) simplifies to \(-i\).
By understanding the cyclical nature of \(i\) and using steps such as the modulo operation, simplifying expressions involving imaginary numbers becomes a systematic and clear process.

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