Problem 38 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{4+\sqrt{-20}}{2} $$

Step-by-Step Solution

Answer

The simplified complex number is \( 2 + \sqrt{5}i \).

Step 1: Simplify the square root of a negative number

The expression \( \sqrt{-20} \) can be rewritten using the imaginary unit, \( i \), where \( i^2 = -1 \). Thus, \( \sqrt{-20} = \sqrt{20} \times \sqrt{-1} = \sqrt{4 \times 5} \times i = 2\sqrt{5}i \).

Step 2: Substitute the simplified imaginary part back into the expression

Replace \( \sqrt{-20} \) in the original expression with \( 2\sqrt{5}i \), giving us: \[ \frac{4 + 2\sqrt{5}i}{2} \].

Step 3: Simplify the fraction

Divide each term in the numerator by 2: \( \frac{4}{2} + \frac{2\sqrt{5}i}{2} = 2 + \sqrt{5}i \).

Step 4: Express the result as a simplified complex number

The expression is now simplified as the complex number \( 2 + \sqrt{5}i \), where 2 is the real part and \( \sqrt{5}i \) is the imaginary part.

Key Concepts

imaginary unit

Complex numbers are a fascinating area of mathematics, and one of their key elements is the imaginary unit, denoted as \(i\). The imaginary unit is defined with the fundamental property that \(i^2 = -1\). This allows us to work with the square roots of negative numbers, which are not defined in the set of real numbers.

To incorporate the imaginary unit in calculations, we rewrite the square root of any negative number, such as \(\sqrt{-20}\), by factoring out \(\sqrt{-1}\). So, \(\sqrt{-20}\) becomes \(\sqrt{20} \times i\). We can further simplify \(\sqrt{20}\) to \(2\sqrt{5}\), leading us to \(\sqrt{-20} = 2\sqrt{5}i\).

Recognizing and properly using the imaginary unit \(i\) is crucial when dealing with complex numbers, as it helps us manage operations that would otherwise be impossible using only real numbers.

simplifying complex expressions

Simplifying complex expressions involves breaking down complicated expressions into more manageable and understandable components. The goal is to represent the expression in its simplest form, typically as a standard complex number \(a + bi\) where \(a\) and \(b\) are real numbers.

In our exercise, we started with \(\frac{4+\sqrt{-20}}{2}\). After converting \(\sqrt{-20}\) to \(2\sqrt{5}i\), we substituted it back into the expression, resulting in \(\frac{4 + 2\sqrt{5}i}{2}\).

• Substitute all\(\ i\) components correctly after finding the simplified form of square roots involving negative numbers.
• Ensure that each part of the fraction is simplified individually by dividing numerators and denominators appropriately.

Through careful simplification, each part of the expression comes together, resulting in the tidy complex number \(2 + \sqrt{5}i\). This simplification process is essential for clarity and accuracy in mathematical communication.

real and imaginary parts

In the world of complex numbers, any complex number is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Understanding these components helps in visualizing and performing operations on complex numbers.

When we look at the final simplified expression \(2 + \sqrt{5}i\) from our exercise, we identify the real part as 2 and the imaginary part as \(\sqrt{5}i\). The real part is a typical real number, while the imaginary part involves the imaginary unit \(i\).

• The real part represents the horizontal component of a complex number when plotted on a plane.
• The imaginary part involves the vertical component and the presence of \(i\).

By clearly distinguishing real and imaginary parts, we can better manipulate and understand complex numbers in mathematical contexts.