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Chapter 2: Problem 10

Find each product and write the result in standard form. $$-8 i(2 i-7)$$

Short Answer

Expert verified
The product \(-8i(2i-7)\) in standard form is \(16 + 56i\).

Step by step solution

01

Distribute \(-8i\) to each term inside the parentheses

Multiply \(-8i\) by each of the terms inside parentheses. That gives us \(-8i \cdot 2i - 8i \cdot 7\).
02

Simplify the obtained expressions

We know that \(i^2 = -1\). Hence, we simplify the expression obtained in Step 1 to \(-16i^2 + 56i = -16*(-1) \ + 56i = 16 + 56i\).
03

Write the answer in standard form

The resulting expression is already in standard form, \(a + bi\). Specifically, it is \(16 + 56i\), where the real part \(a\) is 16 and the imaginary part \(b\) is 56.

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Key Concepts

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

Standard Form
When working with complex numbers, it's essential to express them in a consistent format known as the standard form. The standard form for a complex number is written as \(a + bi\), where:
  • \(a\) is the real part.
  • \(b\) is the imaginary part.
  • \(i\) is the imaginary unit.
In this expression, both \(a\) and \(b\) are real numbers. The complex number \(16 + 56i\) from our example fits perfectly into this form, with 16 as the real part and 56 as the coefficient of the imaginary part. Writing complex numbers in this manner helps in performing calculations and understanding the components separately. It's a fundamental aspect of handling complex numbers efficiently.
Imaginary Unit
The imaginary unit, denoted by \(i\), is a mathematical concept used to simplify the handling of complex numbers. The unit \(i\) is defined by the equation \(i^2 = -1\). This means that the imaginary unit squared equals negative one. This property allows us to deal with square roots of negative numbers, which aren't possible in the realm of real numbers.

  • If \(i^2 = -1\), then \(i = \sqrt{-1}\).
  • This makes calculations involving imaginary numbers consistent with other algebraic operations.
In our solution, understanding that \(i^2 = -1\) was critical to simplifying the expression from \(-16i^2\) to the real number 16.
Distribution in Algebra
Distribution is a crucial algebraic technique used to simplify and break down complex expressions. This method involves multiplying a single term by each term within a parenthesis. In our problem, we distributed \(-8i\) across the terms \(2i - 7\). Here's what happens step-by-step:
  • Multiply \(-8i\) by \(2i\), resulting in \(-16i^2\).
  • Multiply \(-8i\) by \(-7\), resulting in 56i.


Applying distribution allows for breaking down more complicated problems into manageable parts that can be simplified further. Once the distribution was performed, we proceeded to simplify the products using properties of the imaginary unit.
Multiplication of Complex Numbers
Multiplying complex numbers can seem daunting, but it follows a straightforward process similar to regular algebra. To multiply complex numbers, each term must be multiplied individually, respecting the properties of \(i\).

In our example, the multiplication involved two stages:
  • First, multiply the imaginary parts: \(-8i \cdot 2i = -16i^2\).
  • Second, multiply across: \(-8i \cdot -7 = 56i\).
After multiplying, the key is to remember the special property \(i^2 = -1\) to simplify \(-16i^2\) into a real number, adding it to the imaginary result. This process helps transform complex multiplications into a standard form, making them easier to interpret and work with.

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