Problem 17 Write the expression in the form... [FREE SOLUTION]

91影视

Algebra and Trigonometry with Analytic Geometry

Earl Swokowski, Jeffery A. Cole

$Math Studyset 91影视 Explanations$ Math

12 Edition

Chapter 1: Problem 17

Write the expression in the form $a+b i$, where $a$ and $b$ are real numbers. (a) $i^{73}$ (b) $i^{-46}$

Short Answer

Expert verified

(a) 0 + 1i (b) -1 + 0i

Step by step solution

Understanding powers of i

The imaginary unit $i$ is defined such that $i^2 = -1$. It follows that $i^3 = i^2 \cdot i = -i$, $i^4 = i^2 \cdot i^2 = 1$, and $i^5 = i^4 \cdot i = i$. Thus, the powers of $i$ cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then it repeats.

Simplifying i^73

Since the powers of $i$ cycle every 4, we need to find the remainder of 73 when divided by 4 to determine which power corresponds to $i^{73}$. Calculating the remainder: $73 \div 4 = 18$ remainder $1$. Therefore, $i^{73} = i^1 = i$.

Writing i^73 in a+bi form

Since $i^{73} = i$, the expression can be directly written in the form $a + bi$ as $0 + 1i$, where $a = 0$ and $b = 1$.

Simplifying i^{-46}

To simplify $i^{-46}$, first determine $i^{46}$ using the cycle. Calculate $46 \div 4 = 11$ remainder $2$. Therefore, $i^{46} = i^2 = -1$. Using the property $i^{-n} = \frac{1}{i^n}$, we have $i^{-46} = \frac{1}{i^{46}} = \frac{1}{-1} = -1$.

Writing i^{-46} in a+bi form

Since $i^{-46} = -1$, we can write it in the form $a + bi$ as $-1 + 0i$, where $a = -1$ and $b = 0$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Powers of Imaginary Unit

The imaginary unit, represented as $i$, is a fundamental concept in mathematics, especially when dealing with complex numbers. It is defined such that $i^2 = -1$. From this basic definition, we can derive the powers of $i$. Let's take a look:

$i^1 = i$
$i^2 = -1$
$i^3 = i^2 \cdot i = -i$
$i^4 = i^2 \cdot i^2 = 1$
$i^5 = i^4 \cdot i = i$

As you can see, after $i^5$, the cycle repeats with $i$ as the result. Knowing the basic powers helps us simplify more complex expressions involving $i$ by recognizing patterns.

Cyclic Nature of Powers of i

One interesting property of the imaginary unit $i$ is its cyclic nature.

The powers of $i$ cycle every four steps: $i, -1, -i, 1$.
This means that when handling a power of $i$, you can find the equivalent smaller one by noting the remainder when dividing by 4.

For example, to simplify $i^{73}$:

Divide 73 by 4 and get the remainder, which is 1.
This remainder tells us that $i^{73} = i^1 = i$.

Understanding the cyclic nature helps you quickly and easily find equivalent powers, making it easier to work with large exponents.

Imaginary Unit in a+bi Form

Complex numbers are generally expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. Simplifying powers of $i$ can allow us to express results in this form. Let's reflect on our exercise:

Taking $i^{73}$, we found it simplifies to $i$, which is directly written as $0 + 1i$.
Similarly, for $i^{-46}$, recognizing the cycle, we determine it simplifies to $-1$, expressible as $-1 + 0i$.

This form of representation is essential not only in simplifying complex expressions but also in performing operations like addition, subtraction, and multiplication of complex numbers. By expressing complex numbers in $a + bi$ form, we bring a systematic approach to tackling problems involving complex numbers.

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91影视

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. (a) \(i^{73}\) (b) \(i^{-46}\)

Short Answer

Step by step solution

Understanding powers of i

Simplifying i^73

Writing i^73 in a+bi form

Simplifying i^{-46}

Writing i^{-46} in a+bi form

Key Concepts

Powers of Imaginary Unit

Cyclic Nature of Powers of i

Imaginary Unit in a+bi Form

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