Problem 40 - College Algebra

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

Answer

The solution to the exercise is \(-\frac{45}{99} - \frac{i\sqrt{2}}{11}\)

Step 1: Identify Complex Numbers

To start off, you can simplify \(\sqrt{-18}\) into \(3i \sqrt{2}\) where \(i\) denotes the imaginary unit whose square is -1. Therefore, the expression can be re-written as \(\frac{-15-3i\sqrt{2}}{33}.\)

Step 2: Carry out the Division

Now, carry out the division by splitting the fraction into real and imaginary parts: \(-\frac{15}{33} - \frac{3i\sqrt{2}}{33}\).

Step 3: Write Answer in Standard Form

The standard form of a complex number is \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the unit imaginary number. If we simplify our fractions, the final answer could be written as: \(-\frac{45}{99} - \frac{i\sqrt{2}}{11}\)

Key Concepts

imaginary unit

Complex numbers can be tricky, especially when you first start learning about them. One critical component is the imaginary unit, denoted as \( i \). This is not just any number, it holds a special property: \( i^2 = -1 \).
Understanding this property is key to working with complex numbers. You see, any real number squared is always positive - that's what makes the imaginary unit unique. When you deal with square roots of negative numbers, it signifies imaginary solutions.For example, consider \( \sqrt{-1} \). Normally, a square root would not be defined fornegative values in the realm of real numbers. But by moving into the world of complex numbers, we define it as \( i \).
So, when you encounter radicals like \( \sqrt{-18} \), it breaks down to \( 3i\sqrt{2} \), indicating the presence of the imaginary number.

• Imaginary numbers: Born from the need to solve equations involving negative square roots.
• Role of \( i \): Fundamental building block for complex numbers.

standard form

When working with complex numbers, expressing them in standard form can bring clarity.The standard form for a complex number is \( a + bi \), where \( a \) is the real part, and \( b \) is the coefficient of the imaginary part. This makes it easier to perform operations like addition, subtraction, and comparison.In the given example, the number was initially expressed as \( \frac{-15-3i\sqrt{2}}{33} \).By transforming it step-by-step, the expression becomes \( -\frac{45}{99} \) for the real part,and \( -\frac{i\sqrt{2}}{11} \) for the imaginary component. It's now in the clear standard format \( a + bi \).

• Clarity in operations: Simplifies mathematical manipulations.
• Part identification: Easily distinguishes between real and imaginary components.
• Final expression: Helps ensure the complex number is simplified.

simplification of radicals

Simplifying radicals is an important skill when dealing with complex numbers, especially when they involve imaginary units. In our problem, we had \( \sqrt{-18} \), which needed simplification.The standard approach is to break this down step-by-step. First, separate the square root of the negative part and the integer part. This gives you \( \sqrt{-1} \times \sqrt{18} \).Using the imaginary unit \( i \) to handle \( \sqrt{-1} \), it turns into \( i\sqrt{18} \).
Then simplify \( \sqrt{18} \) as a product of perfect squares, \( 3\sqrt{2} \), resulting in \( 3i\sqrt{2} \).
Simplification of radicals makes complex numbers manageable and easier to work with. It translates seemingly complicated expressions into their simplest forms.

• Separating components: \( \sqrt{-18} \rightarrow \sqrt{-1} \times \sqrt{18} \)
• Using \( i \): Converts \( \sqrt{-1} \) into \( i \).
• Simplification: Completes the process, making further calculations straightforward.