Problem 48 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression \( \sqrt{\frac{-9}{4}} \) in the form \( a + bi \) is \( 0 + \frac{3}{2}i \).

Step 1: Separate the Radical into Real and Imaginary Parts

Start by re-writing the original expression \( \sqrt{\frac{-9}{4}} \) as \( \sqrt{-1} \times \sqrt{\frac{9}{4}} \). This is possible because of the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) when dealing with non-negative numbers \, \text{(except when dealing with complex numbers)}.

Step 2: Evaluate \( \sqrt{-1} \)

Recognize that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit. The imaginary unit \( i \) satisfies \( i^2 = -1 \).

Step 3: Evaluate \( \sqrt{\frac{9}{4}} \)

Find the square root of \( \frac{9}{4} \) by taking the square root of the numerator and the denominator separately, which gives us \( \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2} \).

Step 4: Combine the Results

Multiply the results from Step 2 and Step 3 together: \( i \times \frac{3}{2} = \frac{3}{2}i \).

Step 5: Express in Form \( a + bi \)

Write the final answer in the standard form of a complex number. In this expression, there is no real part, so it is \( 0 + \frac{3}{2}i \).

Key Concepts

Imaginary Unit

The imaginary unit is denoted by the letter \(i\). It is a fundamental concept in the field of complex numbers. The essence of \(i\) is to provide a solution for the equation \(x^2 = -1\), which has no solutions among real numbers. Thus, \(i\) is defined such that \(i^2 = -1\). This concept introduces the idea of numbers that involve the square roots of negative numbers. Whenever you encounter the square root of a negative number, \(i\) becomes significant.

• \(i\) squared equals -1: \(i^2 = -1\)
• \(i\) is used to express imaginary parts of complex numbers

Understanding the imaginary unit is crucial for dealing with complex numbers and expressions that result in the square roots of negative values.

Radical Expression

A radical expression involves a root symbol, often indicating a square root or higher roots such as cube roots. In this context, the expression \(\sqrt{\frac{-9}{4}}\) is a radical expression. It represents the square root of the fraction \(\frac{-9}{4}\), which includes a negative sign inside the radical.To simplify a radical expression, you often separate it into real and imaginary components, especially when dealing with a negative number under the square root. This can be done using properties of radicals that allow the separation of multiplication within the root.

• Radical expressions can involve real numbers, complex numbers, or both
• Breaking down radicals helps to isolate imaginary units

The careful handling of radicals with negative values is a stepping stone to understanding complex numbers.

Square Root

The square root of a number \(x\) is a value that, when multiplied by itself, yields \(x\). The square root is denoted as \(\sqrt{x}\).In the context of complex numbers, when you encounter a negative under the square root, the result transforms into an imaginary number. \(\sqrt{-1}\) specifically transforms into the imaginary unit \(i\).

• The square root is fundamental when dealing with both real and complex numbers
• For positive numbers, square roots have two values: a positive and a negative

Understanding square roots is integral for simplifying expressions and handling both real and imaginary components.

Standard Form of a Complex Number

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, and \(bi\) represents the imaginary part of the complex number.The complex number is a combination of both a real and an imaginary part. This form is very useful for mathematical calculations and solving equations involving complex numbers.

• The real part (\(a\)) can be zero, making the entire number purely imaginary, as in \(0 + bi\)
• Both parts of a complex number are crucial for its properties

Using the standard form makes complex numbers easier to visualize and manipulate in equations.