/*! 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 34 Simpify \(\frac{2-7 i}{5+3 i}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 34

Simpify \(\frac{2-7 i}{5+3 i}\)

Short Answer

Expert verified
The simplified expression is \( \frac{31}{16} - \frac{41}{16}i \).

Step by step solution

01

Identify the conjugate of the denominator.

The denominator of the given fraction is \(5+3i\). The conjugate of this expression is found by changing the sign of the imaginary part, making the conjugate \(5-3i\).
02

Multiply the numerator and the denominator by the conjugate.

Our goal is to eliminate the imaginary part from the denominator. We do this by multiplying the numerator and the denominator by the conjugate we found in step 1, which is \(5-3i\): \[\frac{2-7i}{5+3i} \times \frac{5-3i}{5-3i}\]
03

Multiply the numerators.

Multiply the numerators and combine like terms: \[(2-7i)(5-3i) = 2(5) + 2(-3i) - 7i(5) - 7i(-3i)\] \[= 10 - 6i - 35i + 21i^2\] Since \(i^2 = -1\), we can substitute this for the \(i^2\) term: \[= 10 - 6i - 35i - 21(-1)\] \[= 10 - 6i - 35i + 21\] Now combine the real and imaginary terms: \[= (10+21) + (-6i-35i)\] \[= 31 - 41i\]
04

Multiply the denominators.

Now, multiply the denominators and combine like terms: \[(5+3i)(5-3i) = 5(5) + 5(-3i) + 3i(5) -3i(-3i)\] \[= 25 - 15i + 15i + 9i^2\] As mentioned before, substitute \(i^2 = -1\) for the \(i^2\) term: \[= 25 - 15i + 15i - 9(-1)\] \[= 25 - 9\] Now combine the real terms: \[= 16\]
05

Write the simplified fraction.

Now that we have the simplified numerator and denominator, we can write the simplified fraction: \[\frac{31 - 41i}{16}\]
06

Write the final expression in the form a + bi.

To write the final expression in the form \(a + bi\), we can treat the real and imaginary parts separately: \[\frac{31}{16} + \frac{-41i}{16}\] \[= \frac{31}{16} - \frac{41}{16}i\] The simplified expression is: \[\frac{31}{16} - \frac{41}{16}i\]

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

Let \(C^{n}(R)\) denote the set of all real-valued functions defined on the real line that have a continuous \(n\)th derivative. Prove that \(C^{n}(R)\) is a subspace of \(\mathcal{F}(R, R)\).

Show that \(P_{n}(F)\) is generated by \(\left\\{1, x, \ldots, x^{n}\right\\}\).

Let \(V\) be a finite-dimensional vector space over \(C\) with dimension \(n\). Prove that if \(\mathrm{V}\) is now regarded as a vector space over \(R\), then \(\operatorname{dim} \mathrm{V}=\) \(2 n\). (See Examples 11 and 12.)

The set of all \(n \times n\) matrices having trace equal to zero is a subspace \(W\) of \(M_{n \times n}(F)\) (see Example 4 of Section 1.3). Find a basis for W. What is the dimension of W?

Let \(u=(1,2,-2), v=(3,-12,4),\) and \(k=-3\) (a) Find \(\|u\|,\|v\|,\|u+v\|,\|k u\|\) (b) Verify that \(\|k u\|=|k|\|u\|\) and \(\|u+v\| \leq\|u\|+\|v\|\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.