Problem 31 Evaluate the expression and writ... [FREE SOLUTION]

Chapter 1: Problem 31

Evaluate the expression and write the result in the form a bi. $$ i^{3} $$

Short Answer

Expert verified

$i^3 = -i$, or $0 - i$.

Step by step solution

Recall the Powers of i

The imaginary unit $i$ has a cyclic pattern when raised to powers. The first few values are: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This cycle repeats every four powers.

Use the Cyclic Pattern

Based on the cyclic pattern established in Step 1, we can see that $i^3 = -i$. This pattern repeats, so for any integer if you determine the equivalent power in the cycle, you can find the result.

Write the Result in Standard Form

Since the expression $i^3 = -i$, we can rewrite this in the standard form $a + bi$ where $a = 0$ and $b = -1$. Therefore, the expression in the form $a + bi$ is $0 - i$.

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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 represents the square root of -1, something that isn't possible in the realm of real numbers. This is because no real number squared gives a negative result. By using $i$, we can extend our number system to include complex numbers.

The defining property of $i$ is $i^2 = -1$.
Complex numbers take the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Understanding the imaginary unit allows us to perform algebraic operations on complex numbers, providing solutions to equations that do not have real roots.

Powers of i

The powers of the imaginary unit $i$ exhibit a recurring pattern. This happens because $i^2 = -1$. Hence, further powers are derived from multiplying $-1$ with previous results. Here are the first few powers of $i$:

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

These values repeat in a cycle with every four powers. This predictable pattern simplifies calculations involving large powers of $i$ by reducing the power to one within this cycle: mod 4.

Cyclic Pattern of i

The cyclic pattern of $i$ powers is a concept that greatly simplifies working with complex numbers. As observed, the cycle is four steps long: $i, -1, -i, 1$. After four steps, it repeats:

For $i^1$, the result is $i$.
For $i^2$, the result is $-1$.
For $i^3$, the result is $-i$.
For $i^4$, the result is $1$.

Understanding this pattern allows you to quickly determine any power of $i$. If you're asked, for example, to find $i^{99}$, you don't need to compute each power up to 99. Instead, you find $99 \mod 4 = 3$ and use the result of $i^3 = -i$. This cyclic property is crucial for simplifying expressions and problems involving complex numbers.

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91影视

Evaluate the expression and write the result in the form a bi. $$ i^{3} $$

Short Answer

Step by step solution

Recall the Powers of i

Use the Cyclic Pattern

Write the Result in Standard Form

Key Concepts

Imaginary Unit

Powers of i

Cyclic Pattern of i

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