Problem 37 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\frac{-8+\sqrt{-32}}{24}$$

Step-by-Step Solution

Answer

The complex number in standard form is \(-\frac{1}{3} + \frac{i\sqrt{2}}{6}\).

Step 1: Simplify the Square Root

In a complex number, the square root of -1 is represented by \(i\). So, we can break down \(\sqrt{-32}\) into \(\sqrt{-1} * \sqrt{32}\), which is equal to \(i\sqrt{32}\). The square root of 32 can be simplified further into \(4i\sqrt{2}\).

Step 2: Simplify Complex Number

Replace \(\sqrt{-32}\) with \(4i\sqrt{2}\) in the original fraction. This results in \(\frac{-8+4i\sqrt{2}}{24}\).

Step 3: Divide Complex Number

The final step is to divide every term by 24, resulting in \(\frac{-8}{24} + \frac{4i\sqrt{2}}{24}\), which simplifies to \(-\frac{1}{3} + \frac{i\sqrt{2}}{6}\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is crucial when dealing with complex numbers. In mathematics, the imaginary unit is denoted by the symbol \(i\). It is defined as the square root of -1. This definition comes in handy when solving problems involving negative square roots. Without the imaginary unit, handling such roots would be impossible.

Why is it called an 'imaginary' unit? That's because for a long time, numbers involving the square root of negative numbers were thought to be fictitious, thus earning the term 'imaginary.' Despite the name, imaginary numbers are very much real in the world of mathematics, especially in fields like engineering and physics.

In complex numbers, \(i\) serves as a fundamental building block where any number can be expressed in the form \(a + bi\), with \(a\) and \(b\) being real numbers. This form allows easy manipulation in various operations such as addition, subtraction, and multiplication.

Simplifying Radicals

Simplifying radicals is the process of finding an equivalent expression that is easier to manage. In the context of complex numbers, this usually means breaking down square roots of negative numbers using the property of the imaginary unit.

A radical can often be split into two parts: the square root of \(-1\) (i.e., \(i\)), and the square root of a positive number. For example, when simplifying \(\sqrt{-32}\), you break it into \(\sqrt{-1} \times \sqrt{32}\), which translates to \(i \times \sqrt{32}\).

The next step is to simplify the \(\sqrt{32}\) itself. Since 32 can be factored into \(16 \times 2\), where 16 is a perfect square, you can write \(\sqrt{32}\) as \(\sqrt{16} \times \sqrt{2}\), resulting in \(4\sqrt{2}\). Thus, \(\sqrt{-32}\) simplifies to \(4i\sqrt{2}\). This method helps in reducing complex expressions to more manageable ones.

Complex Fraction

Complex fractions can appear intimidating but are manageable with a step-by-step approach. A complex fraction has one or both of its numerator or denominator as algebraic expressions, which could include complex numbers or radicals.

Let's consider the example \(\frac{-8 + 4i\sqrt{2}}{24}\). Here, both the numerator contains a complex number, and the whole forms a complex fraction. To simplify, separate terms to handle them individually, like dividing \(-8\) by 24 and \(4i\sqrt{2}\) by 24.

This results initially in \(\frac{-8}{24} + \frac{4i\sqrt{2}}{24}\), which can further be reduced to fractions with smaller numerators and denominators, ultimately yielding \(-\frac{1}{3} + \frac{i\sqrt{2}}{6}\). Breaking it into these components allows for an organized simplification of complex operations.

Standard Form

The standard form is a way to express complex numbers in a consistent, readable fashion. This standardization makes it easier to compare, operate, and understand complex numbers.

For any complex number, the standard form is \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part. This means we simply rearrange any complex number expression to fit this format.

Consider the given fraction, which is simplified to \(-\frac{1}{3} + \frac{i\sqrt{2}}{6}\). Here, \(-\frac{1}{3}\) is the real part, matched with the imaginary part \(\frac{i\sqrt{2}}{6}\). Expressing it this way makes subsequent calculations straightforward, whether you're adding, subtracting, or multiplying complex numbers. Understanding the standard form helps in seamlessly transitioning back and forth between complex and real number systems.