Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \sqrt{\frac{-9}{4}} = 0 + \frac{3}{2}i \).

Step 1: Rewrite the Expression using Imaginary Numbers

The term \(-9\) inside the square root makes it an imaginary number. Instead of \(\sqrt{-9/4}\), we separate the negative component and express it using the imaginary unit \(i\). Thus, \(\sqrt{-9/4} = \sqrt{\frac{-1 \times 9}{4}} = \sqrt{-1} \times \sqrt{\frac{9}{4}}\). Since \(\sqrt{-1} = i\), we have \(i \times \sqrt{\frac{9}{4}}\).

Step 2: Simplify the Rational Square Root

The expression \(\sqrt{\frac{9}{4}}\) is a rational number. We simplify it by finding the square root of the numerator and the denominator separately, giving \(\frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}\).

Step 3: Multiply by the Imaginary Unit

Combine the results from Step 1 and Step 2. The expression is \(i \times \frac{3}{2}\), which simplifies to \(\frac{3}{2}i\).

Step 4: Express in Standard Form

The standard form of a complex number is \(a + bi\). From the previous calculations, we have the result \(0 + \frac{3}{2}i\), so we express it as \(0 + \frac{3}{2}i\). This makes \(a=0\) and \(b=\frac{3}{2}\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics, allowing us to handle the square roots of negative numbers. When we encounter a square root of a negative number, we use the imaginary unit, denoted as \(i\), where \(i = \sqrt{-1}\). This means that when you take the square of \(i\), you return to the original negative number; \(i^2 = -1\).

• For example, the square root of \(-9\) can be expressed as \(3i\), since \(\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3 \times i\).
• Imaginary numbers are used alongside real numbers to form complex numbers, allowing for more comprehensive mathematical operations and solutions.

Understanding imaginary numbers is crucial as they extend the types of computations and functions we can engage in, especially when arithmetic alone can't solve the problem.

Square Root Simplification

Simplifying square roots involves breaking down complex expressions into simpler parts. In the case of the expression \(\sqrt{\frac{-9}{4}}\), we are dealing with both a negative number and a fraction.
First, realize that the expression is part of an imaginary number because of the negative sign inside the square root. The key step is to separate the imaginary part, using the concept that \(\sqrt{-1} = i\). This allows us to rewrite the expression by isolating \(-1\) and treating it separately from the fraction.

• For example, \(\sqrt{\frac{-9}{4}}\) becomes \(\sqrt{-1} \times \sqrt{\frac{9}{4}}\), which translates into \(i \times \sqrt{\frac{9}{4}}\).
• Next, simplify \(\sqrt{\frac{9}{4}}\) by taking the square root of the numerator and the denominator separately, leading to the simpler form \(\frac{3}{2}\).

By understanding how to break down the process of simplification, we can tackle more complex expressions with confidence.

Standard Form of Complex Numbers

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This format helps standardize calculations and comparisons across mathematical operations.
When we find a result like \(\frac{3}{2}i\), it's expressed in standard form as \(0 + \frac{3}{2}i\). Here, the real part \(a\) is zero, and the coefficient of \(i\) gives \(b = \frac{3}{2}\).

• This form facilitates understanding and working with complex numbers since every number has a clear real and imaginary component.
• It also aids in performing further operations like addition, subtraction, and multiplication of complex numbers.

The standard form ensures that you can easily compare or combine complex numbers while understanding their impact in both real and imaginary dimensions.