Problem 16 Find each product. Express each ... [FREE SOLUTION]

Chapter 7: Problem 16

Find each product. Express each answer in the form $a+b i$ $$4(1-3 i)$$

Short Answer

Expert verified

4 - 12i

Step by step solution

Distribute the constant

Use the distributive property to multiply each term inside the parentheses by 4: \[ 4(1-3i) = 4 \times 1 - 4 \times 3i \]

Perform the multiplications

Calculate the products obtained from distribution: \[ 4 \times 1 = 4 \] \[ 4 \times 3i = 12i \]

Combine the results

Integrate the results to obtain the final expression: \[ 4 - 12i \]

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Key Concepts

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

Distributive Property

The distributive property is a very useful tool in mathematics. It allows us to simplify expressions and make calculations easier. When dealing with multiplication over addition or subtraction, it means we can multiply each term inside a set of parentheses by the factor outside. For example, in our exercise, we apply the distributive property to the expression $ 4(1-3i) $. This means we will multiply 4 by each term inside the parentheses, giving us:
\[ 4 \times 1 - 4 \times 3i \]

This step-by-step approach helps simplify complex expressions and solve them more easily.

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers. Represented by the symbol \i\, it is defined as $ i^2 = -1 $. This definition helps us understand and perform calculations involving complex numbers. In our exercise, we have the term $ -12i $, which includes the imaginary unit $ i $.

Imaginary numbers arise when we deal with the square roots of negative numbers. For example, \sqrt{-9} = 3i\ because $ (3i)^2 = 3i \times 3i = 9i^2 = 9(-1) = -9 $. Understanding the imaginary unit helps us work with complex numbers and solve algebraic equations that have no real solutions.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form $ a+bi $, where \a\ is the real part, and \bi\ is the imaginary part. In our exercise, the result $ 4 - 12i $ is a complex number with a real part of 4 and an imaginary part of \-12i\.

Complex numbers are used in various fields such as engineering, physics, and applied mathematics. They help simplify the representation of oscillations, electrical circuits, and even solutions to certain differential equations. By understanding how to manipulate and multiply complex numbers, you can tackle a wide range of problems involving real and imaginary parts.

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91影视

Find each product. Express each answer in the form \(a+b i\) $$4(1-3 i)$$

Short Answer

Step by step solution

Distribute the constant

Perform the multiplications

Combine the results

Key Concepts

Distributive Property

Imaginary Unit

Complex Numbers

One App. One Place for Learning.

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