Problem 109 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ \frac{6+\sqrt{-18}}{3} $$

Step-by-Step Solution

Answer

The expression \( \frac{6+\sqrt{-18}}{3} \) in the form \( a+bi \) is \( 2+\sqrt{2}i \).

Step 1: Simplify the Square Root of Negative

First, we simplify the expression under the square root: \( \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} \). Since \( \sqrt{-1} = i \), rewrite it as: \( \sqrt{18} \cdot i \). Next, simplify \( \sqrt{18} \) to \( 3\sqrt{2} \). Thus, \( \sqrt{-18} = 3\sqrt{2}i \).

Step 2: Substitute and Simplify the Expression

Substitute \( 3\sqrt{2}i \) back into the expression: \( \frac{6 + 3\sqrt{2}i}{3} \). Now divide both terms in the numerator by 3: \( \frac{6}{3} + \frac{3\sqrt{2}i}{3} \).

Step 3: Perform Division

Simplify each division: \( \frac{6}{3} = 2 \) and \( \frac{3\sqrt{2}i}{3} = \sqrt{2}i \). Therefore, the expression becomes \( 2 + \sqrt{2}i \).

Key Concepts

Imaginary Unit

In the realm of complex numbers, the imaginary unit is a pivotal concept. It is denoted by the symbol \(i\), which stands for the square root of negative one, that is \(\sqrt{-1} = i\). This is a cornerstone idea because in our traditional number system, the square root of a negative number doesn't exist.
To handle such cases, mathematicians introduced the imaginary unit \(i\). With this, we can express square roots of negative numbers in a coherent manner.
This allows us to expand our number system into the complex numbers, which include both real numbers and these new imaginary numbers. The equation \(i^2 = -1\) is crucial for calculations that involve complex numbers and powers of \(i\) to simplify expressions further.

• \(i^0 = 1\)
• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\), and so forth cyclically.

Square Root Simplification

Simplifying square roots, especially those with negative components, is a handy skill when dealing with complex numbers. Let's break down the simplification process:
When you encounter a square root of a negative number like \(\sqrt{-18}\), recognize it as a product of two parts: the real part \(\sqrt{18}\) and the imaginary unit \(\sqrt{-1}\).
Since \(\sqrt{-1} = i\), you rewrite \(\sqrt{-18}\) as \(\sqrt{18} \cdot i\).
Next, simplify \(\sqrt{18}\), which equals \(3\sqrt{2}\). Combining these, \(\sqrt{-18} = 3\sqrt{2}i\). This way, we transform the loosely defined operation into a precise expression using the imaginary unit.
This step is important because it helps convert complex expressions into a standard form that can be more easily understood and manipulated.

Real and Imaginary Parts

In complex numbers, the form \(a + bi\) represents two distinct parts: the real part \(a\) and the imaginary part \(bi\). This structure allows us to express numbers that are combinations of real and imaginary values.
In our problem, after simplification, the expression is rewritten as \(2 + \sqrt{2}i\).
- **Real Part**: The "2" in this expression is the real part.
- **Imaginary Part**: \(\sqrt{2}i\) is the imaginary part, where \(\sqrt{2}\) is the coefficient of \(i\), the imaginary unit.
It's vital to distinguish between these parts because they determine how complex numbers are understood and used in calculations.
When dealing with operations like addition or subtraction, combine the real parts with real parts and imaginary parts with imaginary parts. Only numbers with the same imaginary unit \(i\) can interact directly, which keeps calculations accurate and understandable.