Problem 38 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\frac{-12+\sqrt{-28}}{32}$$

Step-by-Step Solution

Answer

The result is \(-0.375 + 0.08839i\).

Step 1: Identify complex number

Recognize that \(\sqrt{-28}\) is a complex number. We rewrite it as \( \sqrt{-1*28} = \sqrt{-1}*\sqrt{28} = i*\sqrt{28}\). So, the exercise can be rewritten as \(\frac{-12+i*\sqrt{28}}{32}\).

Step 2: Simplify \(\sqrt{28}\)

\(\sqrt{28}\) can be simplified by expressing 28 as a product of its prime factors: \(28 = 2^2*7\). Thus, \(\sqrt{28} = \sqrt{4*7} = 2\sqrt{7}\). Now, the exercise looks like this: \(\frac{-12+2i\sqrt{7}}{32}\).

Step 3: Perform Division

Divide every term inside the numerator by 32: \(-12/32 + (2i\sqrt{7})/32\). This equals to \(-0.375 + 0.08839i\).

Step 4: Write Answer in Standard Form

Now we just need to write the solution in the form \(a+bi\), where \(a\) is the real part, and \(b\) is the imaginary part of the complex number. So, the final answer will be \(-0.375 + 0.08839i\).

Key Concepts

Standard Form of Complex Numbers

A fundamental aspect of complex numbers is their representation in standard form, which looks like \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, and \(i\) is the square root of \-1, the imaginary unit. It's crucial for any calculation involving complex numbers that the resulting number be in this format. This allows for clear communication of both the real and the imaginary components.

In our exercise, the objective is to start with a complex expression and perform operations to render it into standard form. That involves identifying the imaginary number part, simplifying it, dividing complex numbers if needed, and finally expressing the result clearly as \(a + bi\), facilitating comprehension and further calculations.

Complex Number Operations

When performing operations with complex numbers, we adhere to the same arithmetic rules as with real numbers, with the addition of handling the imaginary unit \({i}\). For instance, to add or subtract complex numbers, we simply add or subtract the real parts with each other and the imaginary parts with each other. Multiplication might involve using the FOIL method (First, Outside, Inside, Last) and remembering that \({i^2 = -1}\). In division, we often employ the technique of multiplying by the conjugate to remove the imaginary part from the denominator.

Our textbook exercise illustrates a complex division. After simplification, we divide both parts of the numerator by the denominator separately. This operation is basic yet crucial, as we need to ensure both the real and imaginary components are divided correctly to maintain the accurate representation of the complex number.

Simplifying Square Roots of Negative Numbers

To simplify square roots of negative numbers, we must recognize the role of the imaginary unit \(i\), which is defined as \(\sqrt{-1}\). With this concept, any negative number inside a square root can be expressed as a product of \(i\) and the square root of the positive counterpart of that number. For example, \(\sqrt{-28}\) is rewritten as \(i\sqrt{28}\), breaking it down into a more approachable form.

In our exercise's context, after recognizing the complex number, it's necessary to simplify \(\sqrt{28}\) further. This action involves factorizing 28 and finding its square root in terms of the square roots of its prime factors. Following these steps meticulously allows us to discover simpler representations of complex numbers and provide clear, understandable solutions.