Problem 48 - Algebra and Trigonometry

Question

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \sqrt{\frac{-9}{4}} $$

Step-by-Step Solution

Answer

0 + \frac{3}{2}i

Step 1: Simplify the Fraction Under the Square Root

The expression given is \( \sqrt{\frac{-9}{4}} \). The first step is to separate the fraction into two parts under the square root: \( \sqrt{\frac{-9}{4}} = \sqrt{\frac{9}{4}} \times \sqrt{-1} \).

Step 2: Evaluate the Real Part

Evaluate \( \sqrt{\frac{9}{4}} \). This can be written as \( \frac{\sqrt{9}}{\sqrt{4}} \), which results in \( \frac{3}{2} \).

Step 3: Evaluate the Imaginary Part

The term \( \sqrt{-1} \) is defined as the imaginary unit \( i \). Therefore, \( \sqrt{-1} = i \).

Step 4: Combine the Real and Imaginary Parts

Now combine the results from Step 2 and Step 3: \( \frac{3}{2} \times i \). This means the expression can be written as \( 0 + \frac{3}{2}i \), where the real part is 0 and the imaginary part is \( \frac{3}{2}i \).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept when working with complex numbers. It is denoted by the letter \( i \) and is defined as \( \sqrt{-1} \). This definition allows mathematicians to extend the real number system to include solutions to equations that do not have real solutions. For example, the equation \( x^2 + 1 = 0 \) does not have a solution in the real number system, but using the imaginary unit, we can express its solutions as \( x = i \) and \( x = -i \). In this problem, we encountered the expression \( \sqrt{-1} \), which simplifies directly to \( i \). Understanding the role of the imaginary unit \( i \) is crucial for manipulating expressions involving complex numbers. The presence of \( i \) indicates that the number is not just a real number but one that incorporates an imaginary component.

Radical Expression

A radical expression involves roots, such as square roots, cube roots, etc. In this exercise, we dealt with a square root of a fraction, specifically \( \sqrt{\frac{-9}{4}} \). The presence of a negative number beneath the radical can initially seem puzzling, but by factoring the expression as \( \sqrt{\frac{9}{4}} \times \sqrt{-1} \), we simplify our work by separating the real and imaginary components. When simplifying such expressions:

• Identify separate parts under the square root.
• Simplify each part individually.
• Combine the results to write the final expression.

By breaking down complex radical expressions into simpler parts, we make the process of managing them more intuitive and accessible.

Simplification Process

The simplification process involves reducing an expression to its simplest form. In our exercise, the expression to be simplified was \( \sqrt{\frac{-9}{4}} \). Our goal was to express it in the form \( a + bi \), where \( a \) and \( b \) are real numbers. We approached the simplification by first dividing the square root of the fraction into two manageable parts: a real part \( \sqrt{\frac{9}{4}} \), and the imaginary unit \( \sqrt{-1} \). By evaluating \( \sqrt{\frac{9}{4}} \), we obtained \( \frac{3}{2} \), and recognizing that \( \sqrt{-1} = i \), we assembled the expression as \( 0 + \frac{3}{2}i \). This simplification process enables clearer expression and usage of complex numbers.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. When dealing with complex numbers, we often manipulate algebraic expressions that involve both real and imaginary components. In our example, we worked with an expression involving a square root combined with a fraction. The aim was to express it in the form of \( a + bi \), a standard form for complex numbers. Typically:

• Identify all parts of the expression, both real and imaginary components.
• Simplify each component separately.
• Combine your results to form the final complex expression.

Understanding how to work with algebraic expressions is essential in various fields of mathematics and engineering, as it facilitates the simplification and manipulation of more complex equations and formulas.