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Chapter 10: Problem 59

Multiply. Give answers in standard form. $$ (4+3 i)(1-2 i) $$

Short Answer

Expert verified
10 - 5i.

Step by step solution

01

Apply the distributive property

To multiply (4+3i)(1-2i), use the distributive property: (4)(1) + (4)(-2i) + (3i)(1) + (3i)(-2i).
02

Multiply the real and imaginary parts

Calculate each term separately: - Real part: (4)(1) and (3i)(-2i) - Imaginary part: (4)(-2i) and (3i)(1). So we have: 4 - 8i + 3i - 6i^2
03

Simplify the imaginary terms

Recall that i^2 = -1. Substitute -1 for i^2 in the expression and simplify: 4 - 8i + 3i - 6(-1). This simplifies to: 4 - 8i + 3i + 6.
04

Combine like terms

Combine the real terms and the imaginary terms: The real part is: 4 + 6 = 10. The imaginary part is: -8i + 3i = -5i. So the final answer is: 10 - 5i.

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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 useful technique in algebra. It lets us multiply a single term by two or more terms inside a parenthesis. You may know it from simple arithmetic, where it states:
  • \(a(b + c) = ab + ac\)
Applying this to complex numbers, the rule works similarly. In this example, \((4+3i)(1-2i)\), we'll distribute each term in the first parenthesis by each term in the second. This looks like:
  • (4)(1)
  • (4)(-2i)
  • (3i)(1)
  • (3i)(-2i)
Breaking it down step-by-step ensures that nothing is overlooked, making it easier when dealing with multiplication of complex numbers.
imaginary unit
The imaginary unit is a fundamental element in complex numbers. It is denoted by \(i\), and it has a unique property:
  • \(i^2 = -1\)
Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\). In the context of our problem, notice how we handle the term \(3i(-2i)\). When performing this multiplication, we use the fact that \(i^2 = -1\) to simplify:
  • \(3i(-2i) = 3(-2)(i^2) = 3(-2)(-1) = 6\)
This property helps us handle and simplify expressions involving \(i\) during multiplication accurately.
standard form
The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. To write the final answer in standard form, we combine all the like terms from our expression. Following our example, after using the distributive property and simplifying the imaginary unit: \(4 - 8i + 3i - 6(i^2)\), we simplify:
  • \(4 - 8i + 3i + 6\) as \(i^2 = -1\)
  • Combine the real parts: \(4 + 6 = 10\)
  • Combine the imaginary parts: \(-8i + 3i = -5i\)
So the complex number in standard form is: \(10 - 5i\)

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Most popular questions from this chapter

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