Problem 82 Find each power of i. $$ i^{... [FREE SOLUTION]

Chapter 8: Problem 82

Find each power of i. $$ i^{83} $$

Short Answer

Expert verified

$ i^{83} = -i $

Step by step solution

Understand the Cycle of Powers of i

The powers of the imaginary unit i repeat every four exponents. The basic cycle is: \[ i^1 = i \] \[ i^2 = -1 \] \[ i^3 = -i \] \[ i^4 = 1 \] This cycle then repeats for higher powers.

Determine the Remainder of Exponent by 4

To find $ i^{83} $, determine the exponent鈥檚 remainder when divided by 4. \[ 83 \, \text{mod} \, 4 = 3 \] This means we need to find $ i^3 $.

Use the Cycle to Find the Power of i

From the cycle in Step 1, we know that: \[ i^3 = -i \]

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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, represented by the symbol i, is a mathematical concept used to extend the real number system to the complex number system. The core concept is that i is defined as the square root of -1.

This means:

i^2 = -1

i^1 = i

Using these definitions, mathematicians developed ways to perform arithmetic with complex numbers, which include both a real part and an imaginary part.

Complex numbers can be written in the form: a + bi where a and b are real numbers.

Understanding the imaginary unit i is crucial for dealing with higher mathematical concepts, including those in engineering and physics.

Exponents Modulo

Determining powers of i often involves a technique called exponentiation modulo a number. When evaluating large exponents of i, you don鈥檛 need to perform the exponentiation directly. Instead, you find the exponent modulo 4 because i has a repeating cycle of four.

To compute the value of i raised to a large number (e.g., i^83), you:

1. Divide the exponent by 4.

2. Find the remainder.

In this example, 83 divided by 4 has a remainder of 3, which means: 83 mod4 = 3

This remainder tells you that i^83 is the same as i^3.Understanding how to reduce exponents using modulo is essential because it simplifies the problem significantly, making it much easier to handle mathematically.

Cyclic Patterns in Powers

One fascinating property of the imaginary unit i is its cyclic pattern when raised to increasing powers. This cyclic nature is due to the definition that i^2 = -1.

The cycle is:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

Once you reach i^4, the cycle repeats. Any exponent of i can thus be reduced to one of these four base values.

For example, to find i^83, you only need to find the remainder of 83 divided by 4, which is 3. Therefore, i^83 = i^3 = -iUnderstanding this cyclic pattern helps in simplifying complex problems easily and quickly.
You don't need to continuously multiply i 83 times; knowing the pattern saves a lot of time.

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91影视

Find each power of i. $$ i^{83} $$

Short Answer

Step by step solution

Understand the Cycle of Powers of i

Determine the Remainder of Exponent by 4

Use the Cycle to Find the Power of i

Key Concepts

Imaginary Unit

Exponents Modulo

Cyclic Patterns in Powers

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