/*! 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 60 Find each product. Write the ans... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 60

Find each product. Write the answer in standard form. $$(6-4 i)(6+4 i)$$

Short Answer

Expert verified
The product is 52.

Step by step solution

01

Identify the formula

Recognize that (6 - 4i)(6 + 4i) is a difference of squares formula, which can be written as (a - b)(a + b) = a^2 - b^2.
02

Assign values to a and b

Assign the values a = 6 and b = 4i.
03

Apply the formula

Substitute a = 6 and b = 4i in the difference of squares formula: (6 - 4i)(6 + 4i) = 6^2 - (4i)^2.
04

Calculate the squares

Calculate the squares: 6^2 = 36 and (4i)^2 = 16i^2.
05

Simplify i^2

Remember that i^2 = -1. So, 16i^2 = 16(-1) = -16.
06

Perform the subtraction

Subtract the result: 36 - (-16) = 36 + 16 = 52.

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.

Difference of Squares
The 'difference of squares' is a useful algebraic identity. It states that for any two numbers 'a' and 'b', we have \[ (a - b)(a + b) = a^2 - b^2 \]. Here, 'a' and 'b' can be any values, including complex numbers. When you see an expression like \[ (6 - 4i)(6 + 4i) \], you can recognize it as a difference of squares format. This formula helps to simplify products without expanding the terms step by step.
Multiplication of Complex Numbers
When multiplying complex numbers, use the distributive property. Consider complex numbers in the form \[ (a + bi) \]. To multiply \[ (6-4i) \] by \[ (6+4i) \], you distribute each term: \[ 6(6) + 6(4i) - 4i(6) - 4i(4i) \]. Simplify to: \[ 36 + 24i - 24i - 16i^2 \]. Notice the terms \[ +24i \text{ and } -24i \] cancel out, leaving you with \[ 36 - 16i^2 \]. Next, replace \[ i^2 \] with \[ -1 \], because \[ i^2 = -1 \]. Thus, \[ -16i^2 \] becomes \[ 16 \]. So the result is: \[ 36 + 16 = 52 \]. Using the formula, you avoided complicated steps!
Imaginary Unit
The imaginary unit, denoted as \[ i \], is defined as \[ \sqrt{-1} \]. This number is crucial for dealing with complex numbers. A complex number combines a real part and an imaginary part and takes the form \[ a + bi \], where \[ 'i' \] represents the imaginary unit. One key property is \[ i^2 = -1 \]. Thus, for any power of \[ i \], you can simplify: \[ i^3 = i^2 \times i = -i \]. In problems, it's vital to know \[ i^2 = -1 \] to simplify complex operations. For example, in the expression \[ 16i^2 \], this becomes \[ 16(-1) = -16 \]. Grasping \[ i \]'s properties is essential for working with complex numbers.

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 rational inequality. Write each solution set in interval notation. $$\frac{-4}{1-x}<5$$

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{-5}{i}$$

Solve each equation or inequality. $$|8 x+5|=0$$

Show that \(-3+4 i\) is a solution of the equation \(x^{2}+6 x+25=0\)

Solve each problem. When a model kite was flown in crosswinds in tests to determine its limits of power extraction, it attained speeds of 98 to \(148 \mathrm{ft}\) per sec in winds of 16 to \(26 \mathrm{ft}\) per sec. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.