/*! 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 26 Find each product. Express each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 26

Find each product. Express each answer in the form \(a+b i\) $$(1-3 i)(1+3 i)$$

Short Answer

Expert verified
The product is 10. It can be written as 10 + 岫篿0.

Step by step solution

01

- Apply the distributive property

Combine the like terms
02

- Simplify using the fact that 岫篿^2 = -1

Final answer in the form }a+bi

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.

Distributive Property
The distributive property is a key concept in mathematics, especially when dealing with complex numbers. This property allows us to multiply a single term by each term within a parenthesis. Let's consider the example:
(1 - 3i)(1 + 3i)
To distribute, you multiply each term in the first parenthesis by each term in the second parenthesis. The steps are:
- First, multiply 1 by 1- Second, multiply 1 by 3i- Third, multiply -3i by 1 - Fourth, multiply -3i by 3i
These operations will help us expand the expression and set the foundation for the next steps in solving complex number multiplication.
Combining Like Terms
Once we have applied the distributive property, our next step involves combining like terms. In our example, distributing gives us:
(1 * 1) + (1 * 3i) + (-3i * 1) + (-3i * 3i)
Which simplifies to:
1 + 3i - 3i - 9i^2
Notice there are like terms: +3i and -3i. These terms cancel each other out, so we end up with:
1 - 9i^2
Combining like terms often simplifies the expression, making it easier to solve. This shows how logical and straightforward mathematics can be if we handle each piece step by step.
Complex Conjugate
In our given example,
(1 - 3i)(1 + 3i),
the terms are known as complex conjugates of each other. Complex conjugates have the same real part but opposite imaginary parts. Multiplying complex conjugates will always result in a real number. This is because the imaginary parts cancel out, as demonstrated:
(1 - 3i)(1 + 3i)This occurrence is very useful to simplify complex multiplications and is a fundamental characteristic of complex numbers.
Imaginary Unit Squared
The imaginary unit, represented as i, has a unique property:
i^2 = -1
We used this fact in our previous steps. Once we've combined like terms in our example
(1 + 3i)(1 - 3i)
and obtained1 - 9i^2
we use the fact that
i^2 = -1
So,-9i^2 = -9(-1) = 9
Thus, our expression simplifies to1 + 9 = 10
And our final answer in the form a + bi is10 + 0i
Understanding this unique property of the imaginary unit is crucial for simplifying and solving expressions involving complex numbers effectively.

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 problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Find the length of the side of a square sign whose area is 50 square feet.

Simplify. $$\frac{\sqrt{3}}{\sqrt{2}}+\frac{3 \sqrt{6}}{2}$$

If we use the product rule to simplify \((-1)^{1 / 2} \cdot(-1)^{1 / 2},\) we get $$(-1)^{1 / 2} \cdot(-1)^{1 / 2}=(-1)^{1}=-1$$ If we use the power of a product rule, we get $$(-1)^{1 / 2} \cdot(-1)^{1 / 2}=(-1 \cdot-1)^{1 / 2}=1^{1 / 2}=1$$ Which of these computations is incorrect? Explain your answer.

Work in a small group to write a formula that gives the side of a square in terms of the diagonal of the square and explain the formula to the other groups.

Solve each equation. $$(a+1)^{1 / 3}=-2$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure 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.