Problem 89 Find each power of i. $$ i^{... [FREE SOLUTION]

Chapter 10: Problem 89

Find each power of i. $$ i^{38} $$

Short Answer

Expert verified

i^{38} = -1

Step by step solution

Understanding the Cycle of Powers

The powers of the imaginary unit, i, repeat in a cycle: i^1 = i i^2 = -1 i^3 = -i i^4 = 1This cycle repeats every four exponents. Hence, any power of i can be reduced by recognizing its position in this cycle.

Divide the Exponent by 4

Determine how many complete cycles fit into the exponent and what remains. Compute this by dividing the exponent, 38, by 4 and finding the remainder: 38 梅 4 = 9 remainder 2.

Identify the Equivalent Lower Exponent

Using the remainder, identify the corresponding power of i in the cycle: i^{38} is equivalent to i^2.

Use the Cycle to Find the Value

From the cycle, i^2 = -1. Hence, i^{38} = -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 is a fundamental concept in complex numbers and is denoted by the symbol $i$. The primary property of $i$ is that $i^2 = -1$. This means that $i$ is the square root of -1. Imaginary numbers are essential because they allow the extension of the real number system to solve equations that do not have real solutions, such as $x^2 + 1 = 0$. Learning about $i$ helps in understanding complex numbers and their operations, enabling a range of applications in fields like engineering, physics, and applied mathematics.

Exponent Cycles

One fascinating aspect of the imaginary unit is its exponent cycle. The powers of $i$ repeat in a predictable pattern every four exponents. Here is how the cycle looks:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

After $i^4$, the cycle starts again:

$i^5 = i$
$i^6 = -1$
$i^7 = -i$
$i^8 = 1$

This repeating pattern makes it easy to handle large exponents of $i$. You only need to identify where the exponent fits within this four-step cycle.

Remainder Division

To simplify the powers of $i$, we use remainder division to determine the position of the exponent in the cycle of four.
For example, to find $i^{38}$, you divide 38 by 4 and find the remainder:
$38 \div 4 = 9 \text{ remainder } 2 $.
The remainder, 2, indicates the equivalent lower exponent. This tells us that $i^{38}$ is the same as $i^2$, which is -1.
This process helps simplify high exponentiation into manageable pieces using the cycle.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to a two-dimensional plane using the imaginary unit $i$. A complex number is expressed as $a + bi$, where $a$ and $b$ are real numbers.
In this expression:

$a$ is the real part
$bi$ is the imaginary part

Complex numbers enable the solution of equations that have no real solutions and provide a more comprehensive framework for algebraic operations. They are visualized using the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Knowledge of complex numbers is crucial in many advanced fields of study, including quantum mechanics, signal processing, and control theory.

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91影视

Find each power of i. $$ i^{38} $$

Short Answer

Step by step solution

Understanding the Cycle of Powers

Divide the Exponent by 4

Identify the Equivalent Lower Exponent

Use the Cycle to Find the Value

Key Concepts

Imaginary Unit

Exponent Cycles

Remainder Division

Complex Numbers

One App. One Place for Learning.

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