Problem 81 Find each power of i. $$ i^{... [FREE SOLUTION]

Chapter 8: Problem 81

Find each power of i. $$ i^{43} $$

Short Answer

Expert verified

$i^{43} = -i$

Step by step solution

Understand the powers of i

The powers of the imaginary unit, i, repeat in a cycle of four: oo $$$\begin{aligned} i^1 &= i \ i^2 &= -1 \ i^3 &= -i \ i^4 &= 1 \end{aligned}$$

Determine the remainder when the exponent is divided by 4

Since the powers of i repeat every 4, divide the exponent 43 by 4 and find the remainder to determine where it falls in the cycle. $ \frac{43}{4} = 10$ with a remainder of $3$. Therefore, $ 43 \text{ mod } 4 = 3$.

Identify the corresponding power of i

The remainder from Step 2 is 3. This means that $i^{43}$ is equivalent to $i^3$ since $i^{4k+3} = i^3$ for any integer k.From Step 1, $i^3 = -i$.

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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 a fundamental concept in complex numbers. It is defined as the square root of -1. This might sound confusing at first since the square root of a negative number does not exist in the set of real numbers.
Therefore, mathematicians introduced the imaginary unit to help solve equations that do not have solutions within the real numbers.
Here are some important identities to remember:

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

All higher powers of $i$ can be reduced using these basic identities, helping make complex number calculations easier.

Modulus Operation

The modulus operation, often represented as mod, is a very useful tool in mathematics. It tells you the remainder when one number is divided by another. For instance, when we divide 43 by 4, we get a quotient of 10 and a remainder of 3.
Writing this mathematically, we get:
\ $ 43 \text{ mod } 4 = 3 $
The modulus operation is essential in determining the powers of $i$ because the powers repeat every four. Using the modulus operation, you can reduce any power of $i$ to a more manageable form.
This simplification lets us identify that $i^{43} = i^3 = -i$ quickly.

Cyclic Patterns in Mathematics

Cyclic patterns often appear in different areas of mathematics. As seen with the powers of $i$, they repeat every four steps:

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

After $i^4$, the pattern starts over.
This repetitive sequence helps simplify calculations of higher powers. Other examples of cyclic patterns include:

The last digit of powers of 2: 2, 4, 8, 6
The cycles in trigonometric functions over 2蟺 intervals

Understanding cyclic patterns can save a lot of time and effort in mathematics.
Recognizing these patterns quickly can greatly enhance problem-solving skills and mathematical insight.

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91影视

Find each power of i. $$ i^{43} $$

Short Answer

Step by step solution

Understand the powers of i

Determine the remainder when the exponent is divided by 4

Identify the corresponding power of i

Key Concepts

Imaginary Unit

Modulus Operation

Cyclic Patterns in Mathematics

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