Problem 111 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ \frac{5-\sqrt{-75}}{10} $$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(\frac{1}{2} - \frac{\sqrt{3}}{2}i\).

Step 1: Simplify the Square Root of a Negative Number

Firstly, recognize that when dealing with square roots of negative numbers, you can express them using imaginary numbers. Specifically, \( \sqrt{-75} = \sqrt{75} \times \sqrt{-1} \). Since \( \sqrt{-1} = i \), we can rewrite it as \( \sqrt{-75} = \sqrt{75} i \).

Step 2: Simplify the Square Root of 75

Break down \( \sqrt{75} \) into simpler components. Since 75 is 25 times 3, you have \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \). Thus, \( \sqrt{-75} = 5\sqrt{3} i \).

Step 3: Substitute and Simplify the Complex Expression

Replace \( \sqrt{-75} \) with \( 5\sqrt{3} i \) in the original expression: \[\frac{5 - \sqrt{-75}}{10} = \frac{5 - 5\sqrt{3} i}{10}\]

Step 4: Separate Real and Imaginary Components

Now express the fraction as the sum of its real and imaginary parts in the form \( a + bi \):\[\frac{5 - 5\sqrt{3} i}{10} = \frac{5}{10} - \frac{5\sqrt{3}}{10} i\]

Step 5: Simplify Each Component

Simplify each part of the expression: 1. For the real part: \( \frac{5}{10} \) simplifies to \( \frac{1}{2} \). 2. For the imaginary part: \( \frac{5\sqrt{3}}{10} \) simplifies to \( \frac{\sqrt{3}}{2} \). Thus, the expression becomes \( \frac{1}{2} - \frac{\sqrt{3}}{2} i \).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the world of complex numbers. It's represented by the symbol \(i\) and is defined as the square root of \(-1\). This means that \(i^2 = -1\). Imaginary numbers extend our number system beyond real numbers, thereby enabling the solution of equations that have no real solutions.

• The imaginary unit \(i\) allows us to deal with square roots of negative numbers, which aren't defined in the realm of real numbers.
• When you encounter an expression like \(\sqrt{-a}\), you can rewrite it as \(\sqrt{a} \times \sqrt{-1}\), which simplifies to \(\sqrt{a} i\).

Complex numbers, therefore, are expressed in the form \(a + b i\), where \(a\) is the real part, and \(b\) is the imaginary part. By mastering complex numbers, you expand your mathematical toolbox and gain the ability to solve a wider array of problems.

Square Roots

Square roots are an essential element of both real and complex number systems. A square root essentially refers to a number which, when multiplied by itself, yields the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).To handle square roots of negative numbers:

• Combine the square root of the positive part with the square root of \(-1\), represented by \(i\).
• As seen in \(\sqrt{-75} = \sqrt{75} \times i\), it's split into its real and imaginary components.

To simplify the real portion, factor out perfect squares. For example, \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\). Therefore, \(\sqrt{-75} = 5\sqrt{3} i\). This process reveals both the real and the imaginary contributions within a complex number's structure.

Simplification of Expressions

Simplification of complex expressions involves breaking down each component and consolidating to standard form \(a + bi\). The steps are:

• Separate the real and imaginary parts of the expression from one another. This often involves substitution using \(i\) for the imaginary portion.
• Reduce or simplify all fractions in each component separately, looking for common factors.

In the given expression \(\frac{5 - 5\sqrt{3} i}{10}\):

• The real part \(\frac{5}{10}\) simplifies to \(\frac{1}{2}\).
• The imaginary part \(\frac{5\sqrt{3}}{10} i\) simplifies to \(\frac{\sqrt{3}}{2} i\).

Ultimately, both parts are combined to give a neat form \(\frac{1}{2} - \frac{\sqrt{3}}{2} i\). This streamlined approach ensures clarity and ease in dealing with mathematical operations involving complex numbers.