Problem 42 - Precalculus Student Solutions Manual 5th

Question

Write each number in standard form a \(+b i\) $$\frac{-5+\sqrt{-50}}{10}$$

Step-by-Step Solution

Answer

-\frac{1}{2} + \frac{\sqrt{2}}{2}i

Step 1: Simplify the Square Root

First, recognize that \( \sqrt{-50} \) can be simplified. The term \( \sqrt{-50} \) can be rewritten using the imaginary unit \( i \) as \( \sqrt{50} \cdot i \). Since \( \sqrt{50} = 5 \sqrt{2} \), we have \( \sqrt{-50} = 5 \sqrt{2} \cdot i. \) So the expression inside the numerator becomes \( -5 + 5 \sqrt{2} i \).

Step 2: Separate the Numerator

Rewrite the expression \( \frac{-5 + 5\sqrt{2}i}{10} \) as two separate fractions: \( \frac{-5}{10} + \frac{5\sqrt{2}i}{10} \).

Step 3: Simplify Each Fraction

Simplify each fraction separately. For the first fraction: \( \frac{-5}{10} = -\frac{1}{2} \). For the second fraction: \( \frac{5\sqrt{2}i}{10} = \frac{\sqrt{2}}{2}i \). Combining these results, we get \( -\frac{1}{2} + \frac{\sqrt{2}}{2}i \).

Key Concepts

Standard Form

When working with complex numbers, you often need to convert them into a *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.
The real part, denoted by \( a \), is any real number.
The imaginary part, denoted by \( b \), is multiplied by \( i \).
For example, in the exercise provided, we were given the complex number: \( \frac{-5 + \sqrt{-50}}{10} \).
We converted this into the standard form \( a + bi \) through a series of steps. The final result was \( -\frac{1}{2} + \frac{\sqrt{2}}{2}i \).
Understanding standard form is crucial because it allows us to easily perform operations like addition, subtraction, and multiplication on complex numbers.

Imaginary Unit

The concept of the *imaginary unit* is integral to working with complex numbers. The imaginary unit is denoted by \( i \) and is defined as the square root of -1. That is, \( i = \sqrt{-1} \).
In the given exercise, we encountered \( \sqrt{-50} \). To simplify this, we used the fact that \( i^2 = -1 \), which helps convert the square root of a negative number into a real number multiplied by the imaginary unit, \( i \).
This turns \( \sqrt{-50} \) into \( 5 \sqrt{2}i \).
The imaginary unit \( i \) is what enables us to extend the real number system to include complex numbers, providing a way to handle solutions to equations that would otherwise have no solution within the real numbers alone.

Simplification Process

The *simplification process* is a method used to break down and simplify complex expressions. For the given exercise, the main steps of the simplification process were:

• Simplify the square root of a negative number using \( i \).
• Separate the numerator into distinct fractions.
• Simplify each fraction individually.

For the expression \( \frac{-5 + 5 \sqrt{2} i}{10} \):
First, we rewrote \( \frac{-5 + 5 \sqrt{2} i}{10} \) as two separate fractions: \( \frac{-5}{10} \) and \( \frac{5 \sqrt{2} i}{10} \).
Next, we simplified each fraction: \( \frac{-5}{10} = -\frac{1}{2} \) and \( \frac{5 \sqrt{2} i}{10} = \frac{\sqrt{2}}{2}i \).
Finally, we combined these results to obtain the simplified form \( -\frac{1}{2} + \frac{\sqrt{2}}{2} i \).
This step-by-step approach helps ensure that even complex expressions are broken down into manageable parts, making it easier to arrive at the correct answer.