Problem 90 Simplify. Write each result in a... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 90

Simplify. Write each result in a + bi form. $$\frac{-4}{6 i^{7}}$$

Short Answer

Expert verified

Therefore, the simplified form of the given expression in the form $a + bi$ is $-\frac{2}{3}i$.

Step by step solution

Find the value of $i^{7}$

Note that $i^{7}$ can be rewritten as $i^{4} * i^{3}$. The value of $i^{4}$ is 1 (since $i^{4n}$, where n is any integer, is always 1), and the value of $i^{3}$ is $-i$. Therefore, $i^{7} = 1 * -i = -i.$

Substitute the value of $i^{7}$ into the fraction

Now, let's substitute the value of $i^{7}$ into the given fraction. We have $\frac{-4}{6 i^{7}} = \frac{-4}{6 (-i)}.$

Simplify the fraction

The fraction simplifies to $\frac{-4}{-6i} = \frac{4}{6i} = \frac{2}{3i}$, which is not in the standard form of a complex number $a + bi$. To convert it to the standard form, multiply the numerator and the denominator by the conjugate of the denominator, which is $-i$. Doing so, we get $\frac{2}{3i} * \frac{-i}{-i} = \frac{-2i}{3}$.

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Key Concepts

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

Imaginary unit

Complex numbers are built upon the foundation of the imaginary unit, denoted as $i$. This special component is defined uniquely by the property $i^2 = -1$. When dealing with powers of $i$, it follows a cyclic pattern:

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

This cycle repeats every four terms. So, for any positive integer $n$, you determine the value of $i^n$ by finding the remainder of $n$ divided by 4, and matching it to one of the values above. For example, to find $i^7$, you realize that 7 divided by 4 leaves a remainder of 3, and so, $i^7 = i^3 = -i$. Understanding this cyclical nature helps simplify complex number calculations significantly.

Complex conjugate

The concept of the complex conjugate provides a valuable tool when manipulating complex expressions. Imagine you have a complex number $a + bi$, its complex conjugate is $a - bi$. Essentially, the conjugate is achieved by changing the sign of the imaginary part.

Why is this useful? The primary utility lies in what happens when a complex number is multiplied by its conjugate. The result is always a real number because:

$(a+bi)(a-bi) = a^2 - (bi)^2$
Simplifying further, you know $(-bi)^2 = -b^2(-1) = b^2$
Therefore, the product is $a^2 + b^2$

This real result simplifies calculations such as dividing by a complex number and helps in converting expressions into the more approachable $a + bi$ form.

Simplification

Simplifying complex numbers can seem tricky at first, but it simply requires some straightforward steps. When converting a fraction like $\frac{2}{3i}$ to the standard form $a + bi$, start by eliminating the imaginary unit from the denominator.

You achieve this by multiplying both the numerator and the denominator by the conjugate of the denominator. In this case, with the denominator being $3i$, its conjugate is $-i$.

Multiply: $\frac{2}{3i} \times \frac{-i}{-i} = \frac{-2i}{3i^2}$
Recall $i^2 = -1$, so $3i^2 = 3(-1) = -3$
This means $\frac{-2i}{-3} = \frac{2i}{3}$

The result is $\frac{2i}{3}$, a complex number where the real part, in this case $a$, is zero, and the imaginary part, $b$, is $\frac{2}{3}$. This transformation ensures the fraction is expressed in the simple and familiar form $a + bi$.

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91影视

Simplify. Write each result in a + bi form. $$\frac{-4}{6 i^{7}}$$

Short Answer

Step by step solution

Find the value of \(i^{7}\)

Substitute the value of \(i^{7}\) into the fraction

Simplify the fraction

Key Concepts

Imaginary unit

Complex conjugate

Simplification

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