/*! 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 4 Given \(z=5+2 i,\) find the prod... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Given \(z=5+2 i,\) find the product \(z \cdot \bar{z}\)

Short Answer

Expert verified
29

Step by step solution

01

- Understand the Complex Conjugate

The complex conjugate of a complex number is found by changing the sign of the imaginary part. For the given complex number \(z = 5 + 2i\), its complex conjugate, denoted as \(\bar{z}\), would be \(5 - 2i\).
02

- Write the Expression for the Product

To find the product \(z \times \bar{z}\), write the expression: \((5 + 2i)(5 - 2i)\).
03

- Apply the Difference of Squares Formula

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\). Here, \(a = 5\) and \(b = 2i\). So, the expression becomes \((5)^2 - (2i)^2\).
04

- Calculate Each Part Separately

Calculate \(5^2\) and \((2i)^2\) separately. \(5^2 = 25\) and \((2i)^2 = 4i^2 = 4(-1) = -4\).
05

- Combine the Results

Combine the results from Step 4: \(25 - (-4) = 25 + 4 = 29\).

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!

Key Concepts

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

Complex Conjugate
A complex number is in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate of a complex number switches the sign of the imaginary part. For the complex number \(z = 5 + 2i\), the complex conjugate, denoted as \(\bar{z}\), is \(5 - 2i\). Understanding the complex conjugate is crucial because it helps in various operations such as simplifying expressions and finding magnitudes.
Difference of Squares
The difference of squares formula is given by \((a+b)(a-b) = a^2 - b^2\). This is a valuable tool in algebra and can simplify multiplying binomials. In the case of the problem, we need to find the product \(z \cdot \bar{z}\), where \(z = 5 + 2i\) and \(\bar{z} = 5 - 2i\). By treating \(a\) as 5 and \(b\) as 2i, the product expression \((5 + 2i)(5 - 2i)\) becomes simplified using the difference of squares to find \((5)^2 - (2i)^2\). This not only makes the calculation easier but also utilizes an important algebraic identity.
Imaginary Unit
The imaginary unit \(i\) is defined by the property \(i^2 = -1\). It is the foundation of complex numbers. When squared, it produces a negative value, which plays into our problem. In the given exercise, we encounter \((2i)^2\). Calculating it, we use \(2i \times 2i\), which equals 4\(i^2\). Because \(i^2 = -1\), this is equivalent to 4\(-1\), giving us -4. Remembering that \(i^2 = -1\) is crucial whenever you deal with complex numbers, as this property frequently appears in simplifications and calculations.

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

Solve each equation in the real number system. $$ 2 x^{4}+x^{3}-24 x^{2}+20 x+16=0 $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=x^{4}-5 x^{2}-36 $$

Graph each polynomial function. $$ f(x)=4 x^{4}+7 x^{2}-2 $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=3 x^{4}-3 x^{3}-5 x^{2}+27 x-36 $$

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=4 x^{4}+5 x^{3}+9 x^{2}+10 x+2 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.