Problem 37 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{2+\sqrt{-12}}{2} $$

Step-by-Step Solution

Answer

The simplified complex number is \( 1 + \sqrt{3}i \).

Step 1: Simplify the Square Root

First, we simplify the square root of the negative number. The expression \( \sqrt{-12} \) can be written as \( \sqrt{12} \times \sqrt{-1} \). We know that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit. Therefore, \( \sqrt{-12} = \sqrt{12} \cdot i \).

Step 2: Simplify the Radicand

Next, we simplify \( \sqrt{12} \). We can express 12 as \( 4 \times 3 \), so \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).

Step 3: Substitute Back into the Original Expression

Substitute \( \sqrt{-12} = 2\sqrt{3}i \) back into the original expression: \( \frac{2 + 2\sqrt{3}i}{2} \).

Step 4: Divide Each Term by 2

Now, divide each term in the numerator by 2: \( \frac{2}{2} + \frac{2\sqrt{3}i}{2} = 1 + \sqrt{3}i \).

Step 5: Express as a Simplified Complex Number

The expression \( 1 + \sqrt{3}i \) is now in the form of a complex number \( a + bi \), where \( a = 1 \) and \( b = \sqrt{3} \).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is essential to understanding how to handle square roots of negative numbers. The imaginary unit is denoted as \( i \), and it is defined by the property \( i^2 = -1 \). This definition allows us to simplify square roots of negative numbers in terms of real numbers and \( i \).
When faced with a square root of a negative number, like \( \sqrt{-12} \) in the example, the key is to express it as a product involving \( i \). By relating the square root of the negative number to \( i \), it transforms into something we can handle in our calculations.

• \( \sqrt{-12} \) can be rewritten as \( \sqrt{12} \times \sqrt{-1} \).
• Since \( \sqrt{-1} = i \), this becomes \( \sqrt{12} \cdot i \).

Thus, the imaginary unit \( i \) simplifies the process of dealing with complicated numbers by breaking them into recognizable forms.

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form. This process is crucial when operating with complex numbers because it can streamline calculations.
Take \( \sqrt{12} \), for instance. The goal is to find the prime factorization to make simplification straightforward:

• Find the factors of 12: \( 12 = 4 \times 3 \).
• Recognize that \( 4 \) is a perfect square, allowing \( \sqrt{4} = 2 \).
• Split \( \sqrt{12} \) into \( \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).

By simplifying radicals, you can take the complexity out of calculations, particularly when you're dealing with expressions that include the imaginary unit \( i \). This step is essential because it prepares the expression for further simplification and conversion into a standard complex number form.

Complex Number Operations

Complex number operations involve adding, subtracting, multiplying, and dividing numbers in the form \( a + bi \). To handle operations presented in complex algebra, particularly division, it's necessary to get comfortable with expressing complex numbers clearly.
Starting with the expression \( \frac{2 + 2\sqrt{3}i}{2} \). Each term must be individually simplified, resulting in clearer operations:

• Divide each component by 2: \( \frac{2}{2} + \frac{2\sqrt{3}i}{2} \).
• This simplifies to \( 1 + \sqrt{3}i \).

The resulting expression, \( 1 + \sqrt{3}i \), is in the form \( a + bi \), where \( a = 1 \) and \( b = \sqrt{3} \). Dealing with division in complex numbers often simplifies the process by ensuring each part of the complex expression is handled separately and then combined for the final result.
This proper handling of operations is crucial for mastering complex numbers. It allows you to express results as simplified complex numbers, simplifying communication and error-checking in math.