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Chapter 10: Problem 28

Multiply. Write the product in the form \(a+b i .\) $$ 5 i(4-7 i) $$

Short Answer

Expert verified
The product is \(35 + 20i\).

Step by step solution

01

Distribute the Imaginary Part

Start by distributing the multiplier, which is the imaginary part, across the terms inside the parentheses. We have \[5i imes (4-7i) = 5i imes 4 + 5i imes (-7i).\]
02

Multiply the Real Number

Multiply the first term, which is purely real, by the imaginary unit: \[5i imes 4 = 20i.\]
03

Multiply and Simplify the Imaginary Parts

Now, multiply the second term, which includes the imaginary parts: \[5i imes (-7i) = -35i^2.\] Remember that \(i^2 = -1\), so we have \[-35i^2 = -35(-1) = 35.\]
04

Combine the Real and Imaginary Parts

Combine the results from the previous steps to form a complex number:\[20i + 35.\] Rearrange it to the standard form by changing it to \[35 + 20i.\]

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Key Concepts

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

Imaginary Unit
Complex numbers often involve the term 'imaginary unit', denoted by the letter \(i\). This unit arises from the need to extend the real number system so that we can take square roots of negative numbers. Specifically, \(i\) is defined such that \(i^2 = -1\).
  • The introduction of the imaginary unit allows us to solve equations such as \(x^2 + 1 = 0\), which has no real solutions.
  • The powers of \(i\) cycle in a predictable pattern. For example, \(i^3 = i \times i^2 = i \times (-1) = -i\), and \(i^4 = (i^2)^2 = 1\).
Understanding these properties helps in simplifying expressions and solving equations, especially when multiplying complex numbers.
Distributive Property
The distributive property is a fundamental principle of algebra, applicable to the multiplication of complex numbers. It states that for any three elements, \(a\), \(b\), and \(c\), the expression \(a(b+c)\) can be expanded to \(ab + ac\). This property is crucial when multiplying a complex number by another number.
This exercise uses the distributive property to multiply \(5i\) by each term inside the parentheses:
  • First, multiply \(5i\) by \(4\) (the real number part), resulting in \(20i\).
  • Then, multiply \(5i\) by \(-7i\) (the imaginary part), resulting in \(-35i^2\).
Lastly, we simplify \(i^2\) as \(-1\), turning \(-35i^2\) into \(35\). Using the distributive property ensures each part is accurately considered in the multiplication.
Complex Number Standard Form
The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. This format is useful because it clearly separates the real part and the imaginary part, making calculations and interpretations easier.
  • The real part, \(a\), corresponds to the horizontal axis on the complex plane.
  • The imaginary part, \(b\), is associated with the vertical axis.
In the solution of the exercise, after multiplying, we end up with \(20i + 35\). To write it in standard form, we rearrange it to \(35 + 20i\). This arrangement highlights each component of the complex number, ensuring it meets mathematical conventions and is easier to work with in further calculations.

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