Problem 39 - College Algebra

Question

Perform the indicated operations and write the result in standard form. $$ \frac{-6-\sqrt{-12}}{48} $$

Step-by-Step Solution

Answer

The result in standard form is \(-\frac{1}{8} - \frac{i\sqrt{3}}{24}\).

Step 1: Express the square root of negative number in terms of 'i' square root of positive number.

In this step, it is known that \(\sqrt{-1} = i\) so we can rewrite \(\sqrt{-12}\) as \(\sqrt{-1} \times \sqrt{12}\) which equals \(2i\sqrt{3}\). So, the expression becomes \(\frac{-6-2i\sqrt{3}}{48}\).

Step 2: Simplify the fraction.

The numerator and denominator have common factor of 6. Divide the numerator and the denominator by the common factor to simplify the fraction, so the expression becomes \(\frac{-1-\frac{1}{3}i\sqrt{3}}{8}\).

Step 3: Distribute denominator across the expression to get final result.

Separate the real part and the imaginary part and divide both by 8. Therefore, the expression simplifies to \(-\frac{1}{8} - \frac{i\sqrt{3}}{24}\).

Key Concepts

Imaginary Unit

Complex numbers introduce a fascinating concept in mathematics known as the imaginary unit, denoted as \(i\). The imaginary unit is defined by the property \(i^2 = -1\). This allows us to work with the square roots of negative numbers, which are not possible within the realm of real numbers. For example, \(\sqrt{-1} = i\), and any negative square root can thus be expressed as a multiple of \(i\).

Let's break it down with an example: if you encounter \(\sqrt{-12}\), this can be rewritten as \(\sqrt{-1} \times \sqrt{12}\), which equals \(2i\sqrt{3}\). Here, \(i\) accounts for the \(\sqrt{-1}\), and the rest is treated using normal arithmetic under the square root. These calculations are crucial when simplifying expressions involving complex numbers.

Standard Form

In the realm of complex numbers, expressing the result in standard form is key for clarity and ease of further computations. The standard form is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This format helps us easily identify and handle different parts of a complex number.

Consider the expression \(\frac{-6-2i\sqrt{3}}{48}\). Your goal is to break this down into a form that separates the real and imaginary parts explicitly. After simplifying, we arrive at \(-\frac{1}{8} - \frac{i\sqrt{3}}{24}\). Here, \(-\frac{1}{8}\) is the real part and \(-\frac{i\sqrt{3}}{24}\) is the imaginary part. Keeping complex numbers in this form ensures their properties and potential operations remain clear.

Simplifying Fractions

Fractions are everywhere in math, and simplifying them makes calculations easier and results cleaner. When dealing with complex fractions, the process involves a few steps tailored to handle both real and imaginary parts.

Take the expression \(\frac{-6-2i\sqrt{3}}{48}\). First, identify any common factors in the numerator and the denominator. Here, the common factor is 6. After dividing both by 6, the expression becomes \(\frac{-1-\frac{1}{3}i\sqrt{3}}{8}\).

Another refinement is distributing the denominator across each term, separating real and imaginary parts. So, \(\frac{-1}{8}\) and \(\frac{-i\sqrt{3}}{24}\) clearly identify each portion's contribution, making further manipulation straightforward. Simplifying fractions is a fundamental skill that provides clarity in more complex expressions.