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Chapter 10: Problem 87

Simplify. $$ i^{42} $$

Short Answer

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Step by step solution

01

Understand the powers of i

Recall that the imaginary unit i has the following properties: \[i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1 \] These properties repeat every four powers.
02

Determine the remainder

Since the powers of i repeat every four, find the remainder of 42 divided by 4: \[42 \, \text{mod} \, 4 = 2 \]
03

Simplify using the remainder

The remainder is 2. Thus, \[i^{42} = i^2\]
04

Substitute and finalize

From the properties of i, we know \[i^2 = -1\] Therefore, \[i^{42} = -1\]

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Key Concepts

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

imaginary unit
The imaginary unit, denoted as \( i \), is fundamental in the study of complex numbers. Its defining property is that \( i^2 = -1 \). This creates a new dimension in the number system beyond the real numbers. The concept of the imaginary unit is essential for solving equations that involve the square root of negative numbers. For instance, the square root of -9 is expressed as \( 3i \), because \( 3^2 = 9 \) and thus \( i^2 \) represents the negative part in \( -9 \). The imaginary unit extends the real number system into the complex plane, allowing for a broader range of mathematical operations and solutions.
complex numbers
Complex numbers are numbers that have both a real and an imaginary component. They are written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The real part (\

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