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Chapter 2: Problem 36

Performing Operations with Complex Numbers. Perform the operation and write the result in standard form. $$-8 i(9+4 i)$$

Short Answer

Expert verified
The result of the operation is: 32 - 72i.

Step by step solution

01

Distribute multiplication over addition

Apply distributive law, multiply -8i with both 9 and 4i:\n -8i * 9 + (-8i) * 4i
02

Perform multiplication

Simplify the multiplication:\n -72i + (-32i^2). Here, remember that \(i^2 = -1\) by definition.
03

Substitute \(i^2\) with -1

Substitute \(i^2\) with -1: -72i -32*(-1)
04

Simplify the expression

Now, simplify the expression: -72i + 32.

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Key Concepts

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

Operations with Complex Numbers
Complex numbers are a crucial part of mathematics, especially when dealing with operations that move beyond real numbers. In this exercise, we're working with multiplication involving complex numbers. Remember, a complex number has a real part and an imaginary part, written in the form \(a + bi\).
Operations with complex numbers can include:
  • Adding or subtracting two complex numbers
  • Multiplying or dividing complex numbers
Each operation follows specific rules, similar to those for real numbers, but must also account for the imaginary unit \(i\). It's important to perform these operations without losing sight of the imaginary unit's properties, ensuring terms are simplified correctly and represented in the standard form.
Distributive Property
The distributive property is a fundamental principle in mathematics, stating that multiplying a number by a sum is the same as multiplying the number by each addend and adding those products. It is expressed in the form \(a(b + c) = ab + ac\).
In the context of complex numbers, the distributive property ensures we correctly multiply terms without losing any components along the way. For this exercise, we start by distributing \(-8i\) over the expression \(9 + 4i\) to give us \(-8i \times 9 + (-8i) \times 4i\). This property helps maintain the expression's integrity and ensures each term is considered.
Imaginary Unit
The imaginary unit, represented by \(i\), is a central concept when working with complex numbers. It is defined by the fundamental property that \(i^2 = -1\). This definition allows us to handle calculations involving square roots of negative numbers and is crucial when simplifying results.
In this exercise, after performing the initial multiplication, we obtain \(-32i^2\), which needs simplifying using the key property of the imaginary unit. Replacing \(i^2\) with \(-1\) transforms \(-32i^2\) to \(+32\). This step is critical because it transforms the purely imaginary component into a real number, helping in further simplification.
Standard Form of Complex Numbers
After performing operations, complex numbers should be presented in their standard form, \(a + bi\), where \(a\) and \(b\) are real numbers. This form allows complex numbers to be easily interpreted and compared.
In our exercise, following simplification, we derive the expression \(-72i + 32\). Here, \(32\) is the real part, and \(-72i\) is the imaginary part, placing the final answer neatly into the standard form of complex numbers. Presenting complex numbers in this manner ensures clarity and consistency, especially important in mathematical discussions and calculations.

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