Problem 78 Find each power of i. $$ i^{... [FREE SOLUTION]

Chapter 8: Problem 78

Find each power of i. $$ i^{48} $$

Short Answer

Expert verified

$i^{48} = 1$

Step by step solution

Understand the Powers of i

The imaginary unit i has a cyclical pattern in its powers:- $i^1 = i$- $i^2 = -1$- $i^3 = -i$- $i^4 = 1$These four results then repeat for higher powers of i.

Determine the Remainder

To find $i^{48}$, first determine the remainder when 48 is divided by 4, since the pattern of powers of i repeats every 4 powers.Perform the division: $48 \div 4 = 12$ remainder is 0.

Find the Equivalent Power

Based on the cyclical pattern:- If the remainder is 0, the term corresponds to $i^4$, which is 1.Since $48 \equiv 0 \pmod{4}$, $i^{48}$ is the same as $i^4$.

Solve

Therefore, $i^{48} = 1$.

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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, often denoted as $i$, is a fundamental concept in complex numbers. It is defined as the square root of -1. This might sound strange since no real number squared can produce a negative number. However, in the realm of complex numbers, $i$ makes it possible. Here are some key points to remember:

$i$ is not a real number; it's an imaginary unit.
The primary definition is $i^2 = -1$.
All powers of $i$ can be derived from this definition.

This property allows us to extend the number system and solve equations that have no solutions in the real numbers. For example, the equation $x^2 + 1 = 0$ has no real solutions, but it has two complex solutions: $i$ and $-i$.

cyclical pattern

When working with powers of $i$, you'll notice a repeating pattern. Understanding this cyclical pattern is crucial for simplifying higher powers of $i$. Here's the pattern:

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

Beyond $i^4$, the powers repeat every four steps. This means:

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

And so on. Recognizing this cyclical pattern allows you to quickly determine any power of $i$. For instance, $i^{48}$ simplifies quickly by noting it's a multiple of 4.

remainder

To solve problems involving higher powers of $i$, such as $i^{48}$, use the concept of the remainder. Here's the strategy:

Step 1: Divide the exponent by 4, because the pattern of $i$ repeats every 4 powers.
Step 2: Find the remainder of this division. The remainder indicates which part of the cycle the power corresponds to.
For $i^{48}$:

Divide 48 by 4: $48 \div 4 = 12$ with a remainder of 0.
A remainder of 0 tells us that $i^{48}$ is equivalent to $i^4$.
We know from the cyclical pattern that $i^4 = 1$.

So, $i^{48} = 1$. This method simplifies evaluating any power of $i$ efficiently. Remember, identifying the remainder helps you map the problem back to a simple base power within the cycle.

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91影视

Find each power of i. $$ i^{48} $$

Short Answer

Step by step solution

Understand the Powers of i

Determine the Remainder

Find the Equivalent Power

Solve

Key Concepts

imaginary unit

cyclical pattern

remainder

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