Problem 57 Multiply. Give answers in standa... [FREE SOLUTION]

Chapter 10: Problem 57

Multiply. Give answers in standard form. $$ 5 i(-6+2 i) $$

Short Answer

Expert verified

-10 - 30i

Step by step solution

- Distribute the Imaginary Unit

To multiply, distribute the imaginary unit, 5i, to both terms inside the parentheses: \[ 5i \times (-6) + 5i \times 2i \]

- Multiply Each Term

Perform the multiplication for each term separately: \[ 5i \times (-6) = -30i \] and \[ 5i \times 2i = 10i^2 \]

- Simplify the Imaginary Unit Squared

Recall that $ i^2 = -1 $. Hence, \[ 10i^2 = 10 \times (-1) = -10 \]

- Combine Like Terms

Combine the real part and the imaginary part to express in standard form: \[ -10 - 30i \]

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.

Imaginary Unit

The imaginary unit, commonly represented as $ i $, is a fundamental concept in complex numbers. The key property of $ i $ is that when it is squared, it equals -1, i.e., $ i^2 = -1 $. This property is used to simplify expressions involving complex numbers.

When multiplying complex numbers, you need to remember this property to convert terms involving $ i^2 $ into real numbers. For example, in the exercise, we had $ 5i \times 2i = 10i^2 $. Using the property $ i^2 = -1 $, this simplifies to $ 10 \times -1 = -10 $.

Understanding the imaginary unit is crucial for manipulating and simplifying complex number expressions.

Distributive Property

The distributive property is a crucial algebraic rule that applies to complex numbers multiplication. It states that for any numbers $ a, b, $ and $ c $, the equation $ a(b + c) = ab + ac $ holds. This property allows us to distribute a factor across terms within parentheses.

In the given exercise, we used the distributive property to multiply $ 5i $ with each term inside $(-6 + 2i)$:

$ 5i \times (-6) = -30i $
$ 5i \times 2i = 10i^2 $

By distributing $ 5i $ to both terms inside the parentheses, we break down the problem into simpler parts that can be easily managed.

Standard Form

The standard form of a complex number is written as $ a + bi $, where $ a $ and $ b $ are real numbers, and $ i $ is the imaginary unit. It鈥檚 important to write your final answer in this form because it clearly separates the real and imaginary parts.

In our exercise, after distributing and simplifying, we ended up with $ -10 - 30i $. Here, $ -10 $ is the real part, and $ -30i $ is the imaginary part. Writing the solution in standard form helps in understanding and further processing complex numbers.

Always aim to express your final answer in this form for clarity and consistency.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Multiply. Give answers in standard form. $$ 5 i(-6+2 i) $$

Short Answer

Step by step solution

- Distribute the Imaginary Unit

- Multiply Each Term

- Simplify the Imaginary Unit Squared

- Combine Like Terms

Key Concepts

Imaginary Unit

Distributive Property

Standard Form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.