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Chapter 10: Problem 46

Multiply. $$ (-32 i)(-2 i) $$

Short Answer

Expert verified
-64

Step by step solution

01

Identify the Components

Recognize the components in the expression: \(-32i\) and \(-2i\). Both are complex numbers with imaginary parts.
02

Multiply the Coefficients

Multiply the coefficients of the imaginary parts: \(-32 \times -2 = 64\).
03

Multiply the Imaginary Units

Since \(i \times i = i^2 = -1\), we multiply the imaginary units: \(-32i \times -2i = 64(i^2)\).
04

Simplify the Result

Substitute \(i^2\) with \(-1\) and simplify: \64(i^2) = 64(-1) = -64\.

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Key Concepts

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

Imaginary Unit
When dealing with complex numbers, understanding the imaginary unit is crucial. The imaginary unit is represented by the symbol \(i\). It is defined by the equation:
\[i^2 = -1\]
This means that \(i\) itself is the square root of -1. Complex numbers often appear in the form of \(a + bi\), where \(a\) and \(b\) are real numbers.
In our given problem, \(-32i\) and \(-2i\) are examples of complex numbers, though they only have imaginary parts.
Coefficients
Coefficients are the numerical factors before the imaginary unit \(i\) in complex numbers. In our problem, the coefficients are -32 and -2. These coefficients need to be multiplied first, separate from the \(i\).
In step-by-step terms:
  • Multiply the coefficients: \(-32 \times -2 = 64\).
  • This gives us part of our product without considering the imaginary unit yet.
i Squared
\(i^2\) is a fundamental concept in complex number arithmetic. Remember that \(i^2 = -1\). This property is essential when performing operations involving multiple imaginary units. In the problem:
  • We first recognize that multiplying the imaginary units involves computing \(i \times i\).
  • This results in \(i^2\).
Substituting \(i^2\) with -1 simplifies our expression. Therefore:
\(64(i^2) = 64 \times -1 = -64\).
Understanding and using \(i^2 = -1\) is key to solving such problems correctly.

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Most popular questions from this chapter

Solve each formula for the indicated variable. $$ r=\sqrt{\frac{\mathscr{A}}{\pi}} \text { for } \mathscr{A} $$

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