Problem 40

Question

In Exercises $29-44$, perform the indicated operations and write the result in standard form. $$ \frac{-15-\sqrt{-18}}{33} $$

Step-by-Step Solution

Verified

Answer

The result in standard form is -$\frac{5}{11} - \frac{\sqrt{2}}{11}i$.

1Step 1: Recognize and write the complex number in standard form

Recognize that under the square root is negative, which implies that we have a complex number. The square root of -18 is $i \sqrt{18}$ or $3i\sqrt{2}$, considering that $i^2 = -1$. Therefore, $\sqrt{-18} = 3\sqrt{2} i$. This means the expression changes to $\frac{-15-3\sqrt{2} i}{33}$.

2Step 2: Divide real and complex parts by the denominator

Now we should divide the real part (-15) and the complex part (-3$\sqrt{2}$i) by the denominator (33). This results in $-\frac{15}{33}$ and $-\frac{3\sqrt{2}}{33}i$. Simplifying these fractions, we get $-\frac{5}{11}$ and $-\frac{\sqrt{2}}{11}i$.

3Step 3: Write the final result in standard form

In the previous step, we obtained the simplified real and complex parts of the expression. In this step, we combine them to write the final result in standard form, which is: $-\frac{5}{11} -\frac{\sqrt{2}}{11}i$.

Key Concepts

Complex NumbersSimplify Complex FractionsImaginary Unit $i$

Complex Numbers

Complex numbers are a fundamental concept in mathematics that extend the idea of the one-dimensional number line to a two-dimensional complex plane. A complex number is of the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit, satisfying the equation $i^2 = -1$. The real numbers are a subset of complex numbers where the imaginary part is zero.

When dealing with complex numbers, operations like addition, subtraction, multiplication, and division follow specific rules to ensure consistency across the complex plane. For example, to add or subtract complex numbers, you combine the real parts with the real parts and the imaginary parts with the imaginary parts separately. When you multiply complex numbers, the use of the distributive property and the definition of $i^2 = -1$ become crucial. Complex number division often involves simplifying complex fractions, which may require conjugation or other techniques to eliminate the imaginary unit from the denominator.

Simplify Complex Fractions

Simplifying complex fractions involves reducing the complex number in the numerator and denominator to its simplest form. This often means eliminating the imaginary unit $i$ from the denominator and simplifying the real number components. The goal is to express the result in standard form $(a + bi)$ where $a$ and $b$ are simplified fractions or integers, without a complex denominator.

Here's how you tackle the simplification process:

First, identify and isolate the real and imaginary parts of the complex fraction.
Divide both parts by the common denominator to simplify the fraction.
If the denominator contains a complex number, multiplication by its conjugate might be necessary. This method gets rid of the imaginary unit from the denominator.
Simplify the new real and imaginary components to their lowest terms.

Following these steps will give you a clearer and more manageable version of the complex number, as seen in the textbook exercise.

Imaginary Unit $i$

The imaginary unit $i$ is the backbone of complex numbers, providing a solution to square roots of negative numbers, which don't exist within the realm of real numbers. It is defined such that $i^2 = -1$, and it extends the number system to include numbers that can be used in a variety of mathematical and engineering applications.

Some key points to remember about the imaginary unit include:

$i$ represents the square root of -1.
Even powers of $i$ result in real numbers (for example, $i^2 = -1$, $i^4 = 1$), while odd powers are imaginary.
When simplifying square roots of negative numbers, factor out $i$ to manage the expression as a complex number (for instance, $\sqrt{-18} = i \sqrt{18} = 3i\sqrt{2}$).
Imaginary numbers allow for operations and solutions in mathematics that real numbers alone cannot provide.

The role of $i$ in complex calculations is illustrated in the exercise, where the square root of a negative number transforms a seemingly impossible operation into a manageable one with a real and an imaginary component.

Problem 40