/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Write the quotient in standard f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 59

Write the quotient in standard form. $$\frac{3 i}{(4-5 i)^{2}}$$

Short Answer

Expert verified
The standard form of the quotient \(\frac{3 i}{(4-5 i)^{2}}\) is \(1.48+1.52i\).

Step by step solution

01

Expand the Denominator

The first step is to expand the denominator \((4-5 i)^{2}\). This is done using the square of binomial formula: \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Thus, \((4-5 i)^{2} = 4^{2} - 2(4)(-5i) + (-5i)^{2} = 16 + 40i - 25i^{2}\). Note that \(i^{2} = -1\), so the expression simplifies to \(16 + 40i + 25 = 41 + 40i\).
02

Multiply by the Conjugate

In order to get rid of \(i\) in the denominator, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(41 + 40i\) is \(41 - 40i\). So, \(\frac{3i}{41 + 40i} * \frac{41 - 40i}{41 - 40i} = \frac{3i(41 - 40i)}{41^{2} - (40i)^{2}} = \frac{123i - 120i^{2}}{1681 - 1600} = \frac{123i - 120(-1)}{81} = \frac{123i + 120}{81}\).
03

Write the Result in Standard Form

Now, divide each term in \(\frac{123i + 120}{81}\) by the denominator to put it into the standard form \(a + bi\). The result is \(\frac{120}{81} + \frac{123}{81}i\), which simplifies to \(1.48 + 1.52i\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the given zero to find all the zeros of the function. Function \(g(x)=x^{3}-7 x^{2}-x+87\) Zero \(5+2 i\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{4}+6 x^{3}+10 x^{2}+6 x+9$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(s)=2 s^{3}-5 s^{2}+12 s-5$$

Use the given zero to find all the zeros of the function. Function \(h(x)=3 x^{3}-4 x^{2}+8 x+8\) Zero \(1-\sqrt{3} i\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}+9 x^{2}+27 x+35$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.