Problem 33 Perform the operation and write ... [FREE SOLUTION]

91影视

Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 33

Perform the operation and write the result in standard form. $$(3 i)(-8 i)$$

Short Answer

Expert verified

The result of the operation $(3i) \cdot (-8i)$ in standard form is 24.

Step by step solution

Multiplication

Start by simply multiplying the two complex numbers together just as one would do with real numbers. This would lead us to $(3i) \cdot (-8i)$.

Substitute $i^2$ with -1

Since $i \cdot i = i^2 = -1$, the result from Step 1 can be rewritten as $-24 \cdot -1$.

Simplify

After substituting $i^2$ with -1, simplify the equation to receive the final answer of 24.

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Key Concepts

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

Standard Form

Complex numbers provide a way to represent quantities that have both a real part and an imaginary part. The standard form of a complex number is expressed as $a + bi$, where $a$ represents the real part and $bi$ denotes the imaginary part, with $i$ being the imaginary unit. The beauty of the standard form lies in its simplicity, making it easy to add, subtract, multiply, and divide complex numbers just like real numbers. Understanding the standard form is crucial as it serves as the foundation for performing algebraic operations with complex numbers.

Imaginary Unit

The concept of the imaginary unit, represented by $i$, is pivotal in understanding complex numbers. It's defined as the square root of -1, that is $i = \sqrt{-1}$, which doesn't have a solution in the realm of real numbers. This simple yet profound idea extends the number system beyond the reals, allowing for the representation of numbers that can't be plotted on the standard number line. Multiplying $i$ by itself flips the sign due to its definition: $i^2 = -1$, and knowing this property is essential when simplifying expressions involving complex numbers.

Simplifying Complex Numbers

Simplifying complex numbers often involves applying basic algebraic techniques while keeping in mind the unique properties of the imaginary unit. When multiplying complex numbers, it's common to encounter terms like $i^2$, which can be immediately replaced with -1 to facilitate further simplification. The goal is to break down expressions into their simplest form in standard notation, $a + bi$, where any occurrence of $i^2$ has been taken care of. Simplification aids in interpreting and solving complex-number problems efficiently.

Algebraic Operations

Algebraic operations with complex numbers follow the same fundamental principles as operations with real numbers. Whether it's addition, subtraction, multiplication, or division, the steps are quite similar but require an extra layer of care due to the presence of the imaginary unit. Multiplying complex numbers, for example, means distributing each part of one complex number to each part of the other, applying the FOIL method (First, Outside, Inside, Last), and then simplifying the result. It's a seamless process that, when executed correctly, keeps the beauty and complexity of these numbers intact.

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91影视

Perform the operation and write the result in standard form. $$(3 i)(-8 i)$$

Short Answer

Step by step solution

Multiplication

Substitute \(i^2\) with -1

Simplify

Key Concepts

Standard Form

Imaginary Unit

Simplifying Complex Numbers

Algebraic Operations

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