Problem 39 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\frac{-6-\sqrt{-12}}{48}$$

Step-by-Step Solution

Answer

The simplified form of the given complex fraction is \(-\frac{1}{8} -\frac{\sqrt{3}}{24}i\).

Step 1: Simplify the Square Root

Take out the imaginary unit (\(i\)) from the square root as \(\sqrt{-1} = i\) and that leaves \(\sqrt{12}\).The representation of \(\sqrt{-12}\) then becomes \(2\sqrt{3}i\). Thus the give expression changes to \(\frac{-6-2\sqrt{3}i}{48}\).

Step 2: Simplify the Fractions

Divide both \(-6\) and \(2\sqrt{3}i\) by \(48\). This results to \(\frac{-1}{8}-\frac{\sqrt{3}}{24}i\).

Step 3: Place in Standard Form

The final answer is the standard form of a complex number, given by \(a + bi\). In this case, \(a = -\frac{1}{8}\) and \(b = -\frac{\sqrt{3}}{24}\). So, the answer in standard form is \(-\frac{1}{8} -\frac{\sqrt{3}}{24}i\).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a cornerstone concept. We denote it as \(i\), where \(i\) is defined as the square root of \(-1\). While real numbers are usually located on a straight number line, complex numbers combine real and imaginary parts, allowing us to describe numbers more fully in a two-dimensional space.

Here, it’s critical to understand that the imaginary unit helps us manage square roots of negative numbers, which don't have real solutions. For example, the square root of \(-4\) would be \(2i\) because when squared it gives back \(-4\). This concept opens up the ability to solve real-world problems that involve oscillations, waves, and electronics, which may not have solutions within the realm of real numbers alone.

Standard Form

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part of the number. This form is useful because it succinctly captures both components of a complex number in a single expression.

To write a complex number in standard form, ensure that both the real and imaginary parts are explicitly stated. For example, if given \(-6 - \sqrt{-12}\), you should work to express it as \(a + bi\). This involves simplifying the expression to separate real and imaginary components and divide them appropriately where necessary. This form, \(-\frac{1}{8} - \frac{\sqrt{3}}{24}i\), highlights the balance and structure of complex numbers, making them easier to read and interpret.

Square Root Simplification

Simplifying the square root of a negative number involves using the imaginary unit. The process begins by recognizing that \(\sqrt{-n} = \sqrt{n}i\). Take the number \(-12\) for instance. We rewrite it as \(\sqrt{-1} \times \sqrt{12}\), which translates to \(i\times \sqrt{12}\).

Let's dive into simplifying \(\sqrt{12}\) on its own. Break it down into prime factors: \(12 = 2 \times 2 \times 3\), so \(\sqrt{12} = \sqrt{2^2 \times 3} = 2\sqrt{3}\).

Thus, \(\sqrt{-12} = i \times 2\sqrt{3}\) or simply \(2\sqrt{3}i\). Simplifying the square root this way allows us to efficiently incorporate it into expressions and ultimately write the number in its standard complex form. This technique is very handy because it transforms seemingly complicated expressions into clear and workable parts.