Problem 96 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression is \(2 + \sqrt{2}i\).

Step 1: Simplify the Square Root

The given expression is \( \frac{6 + \sqrt{-18}}{3} \). To simplify, we first find \( \sqrt{-18} \). Remember that \( \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18}i \). Since \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \), we have \( \sqrt{-18} = 3\sqrt{2}i \). Thus, the expression becomes \( \frac{6 + 3\sqrt{2}i}{3} \).

Step 2: Simplify the Fraction

Now, we divide each term in the numerator by 3. Thus, \( \frac{6}{3} = 2 \) and \( \frac{3\sqrt{2}i}{3} = \sqrt{2}i \). Therefore, the expression simplifies to \( 2 + \sqrt{2}i \).

Step 3: Write in the Form \(a + bi\)

The expression \( 2 + \sqrt{2}i \) is already in the correct form \( a + bi \), where \( a = 2 \) and \( b = \sqrt{2} \).

Key Concepts

Imaginary Numbers

Imaginary numbers form an essential extension of the real number system. They arise when we deal with the square roots of negative numbers, something that is not possible with real numbers alone. The imaginary unit is denoted by "i," where \(i\) is defined as the square root of -1. This definition allows us to express any negative square root as a real number multiplied by \(i\).

Let's take the square root of -18 as an example from our exercise. We rewrite it using the imaginary unit as \(\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18}i\). This converts the square root of a negative number into a product of a real number and the imaginary unit.

In general, imaginary numbers are not confined to being mere abstract concepts. They play vital roles in diverse areas of mathematics and engineering. In summary, understanding imaginary numbers is key to mastering complex numbers.

Simplifying Expressions

Simplifying expressions involving complex numbers involves a step-by-step breakdown of the components. Consider our exercise expression \(\frac{6 + \sqrt{-18}}{3}\). We started by simplifying \(\sqrt{-18}\) as explained earlier, which results in \(3\sqrt{2}i\).

The next step involves breaking down the fraction. This is done by dividing each term in the numerator by the number in the denominator separately. So, we divide both 6 and \(3\sqrt{2}i\) by 3, which simplifies the expression to \(2 + \sqrt{2}i\).

Key points when simplifying expressions:

• Break expressions into simpler parts.
• Use basic division principles.
• Keep track of imaginary and real components.

Remember, our goal with simplifying is to make expressions more digestible while preparing them for practical applications or further operations.

Square Roots of Negative Numbers

Regular square roots appear quite straightforward until we face negative numbers. Here, we dive into the idea of square roots of negative numbers, which leads us to embrace the concept of imaginary numbers. Negative numbers do not have real square roots because the square of any real number is non-negative.

Thus, when tasked with finding \(\sqrt{-18}\), we pivot to using the imaginary unit. We rewrite the square root of a negative number as the product of \(\sqrt{-1}\), which is \(i\), and the square root of the positive version of the number. This gives us \(\sqrt{-18} = \sqrt{18} \cdot i\).

Quick tips when dealing with square roots of negative numbers:

• Identify the imaginary unit \(i\).
• Convert negative square roots using \(i\).
• Simplify the positive part first before attaching \(i\).

Learning to approach square roots of negative numbers with comfort helps in broadening understanding in complex number operations.