Problem 104 Write in the form $a+b i$. $... [FREE SOLUTION]

Chapter 7: Problem 104

Write in the form $a+b i$. $$ i^{8}-i^{7} $$

Short Answer

Expert verified

The expression simplifies to $1 + i$.

Step by step solution

Calculate Powers of i

Recall the powers of the imaginary unit $i$: - $i^1 = i$- $i^2 = -1$- $i^3 = -i$- $i^4 = 1$, and the cycle repeats every four powers.Using this cycle, calculate $i^8$ and $i^7$.

Calculate i^8

Since the powers of $i$ repeat every four terms, $i^8$ is equivalent to $(i^4)^2 = 1^2 = 1$.

Calculate i^7

Similarly, $i^7 = i^{4+3} = i^4 \cdot i^3 = 1 \cdot (-i) = -i$.

Subtract the Imaginary Numbers

Subtract $i^7$ from $i^8$ to get the expression in the form $a + bi$:$i^8 - i^7 = 1 - (-i) = 1 + i$.

Present the Result

The expression $i^8 - i^7$ simplifies to $1 + i$, which is in the form $a+bi$ with $a=1$ and $b=1$.

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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, denoted as $i$, is a fundamental concept in complex numbers. It's defined as the square root of $-1$. This might seem unusual because there is no real number whose square gives $-1$. That's where $i$ comes in handy, providing a way to work with squares of negative numbers.
The essential property of the imaginary unit is:

$i^2 = -1$

Understanding $i$ is crucial as it allows us to expand the number system from real numbers to complex numbers, making it possible to solve equations that have no real solutions.

Powers of i

To master complex numbers, you need to understand how the powers of $i$ cycle every four terms. Memorizing the pattern can simplify numerous calculations.

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

Then it repeats: $i^5 = i$, $i^6 = -1$, and so on. Knowing this cycle allows you to effortlessly find higher powers, like in the given problem where $i^8$ and $i^7$ are evaluated. Recognizing these cycles helps reduce complex expressions involving powers of $i$.

Expressions in a+bi Form

The expression $a+bi$ is the standard form for complex numbers. Here, $a$ and $b$ are real numbers, where $a$ is the real part, and $bi$ is the imaginary part of the complex number. This form enables easy arithmetic and visualization on the complex plane.
Given the expression $i^8 - i^7$, it simplifies to $1 + i$ in the $a+bi$ form. By calculating the powers first and subtracting, the expression converts seamlessly into the standard complex number format. Remember, always express your final answer in $a+bi$ when dealing with complex numbers to make the results clear and structured.

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91影视

Write in the form \(a+b i\). $$ i^{8}-i^{7} $$

Short Answer

Step by step solution

Calculate Powers of i

Calculate i^8

Calculate i^7

Subtract the Imaginary Numbers

Present the Result

Key Concepts

Imaginary Unit

Powers of i

Expressions in a+bi Form

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