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Chapter 7: Problem 56

Multiply. Give answers in standard form. $$(-32 i)(-2 i)$$

Short Answer

Expert verified
-64

Step by step solution

01

- Multiply the constants

First, multiply the numeric constants from each term: \(-32 \times -2 = 64\)
02

- Multiply the imaginary units

Next, multiply the imaginary units: \(i \times i = i^2\)
03

- Simplify using the value of i^2

Recall that \(i^2 = -1\). Replace \(i^2\) with \(-1\): \(64 \times (-1) = -64\)

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Key Concepts

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

imaginary unit
The imaginary unit, often represented as \( i \), is a fundamental component in the field of complex numbers. Its defining property is that \( i^2 = -1 \). Understanding this concept is essential in dealing with complex number operations.
For example, when multiplying complex numbers, it helps to remember that \( i \) behaves differently compared to real numbers. If you multiply \( i \) by itself, it transforms into -1. This property will come into play when simplifying expressions involving complex numbers.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. In the context of complex numbers, they can include the imaginary unit \( i \) alongside real numbers.
When multiplying algebraic expressions that involve complex numbers, follow these steps:
  • First, handle the numerical coefficients (or constants) separately.
  • Next, deal with the imaginary unit \( i \) according to its properties.
  • Finally, combine and simplify the results to arrive at the final expression.

For example, in the given exercise, we multiply the constants: \(-32 \times -2 = 64\). Then we multiply the imaginary units: \( i \times i = i^2 \). Lastly, using the known value of \( i^2 = -1 \), we simplify to get \( 64 \times (-1) = -64 \).
standard form
In complex numbers, standard form refers to expressing the number as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
Standard form makes it easier to understand and manipulate complex numbers by separating the real part from the imaginary part.
However, when dealing solely with imaginary numbers (such as the given exercise), the result might be purely real or purely imaginary after simplification.
In our exercise, we multiplied \( -32i \) and \( -2i \). The product was simplified to the real number \( -64 \), which is already in standard form as \( -64 + 0i \). Thus, our final answer is just \( -64 \) in its simplest form.

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