Problem 59 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-24}}{\sqrt{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2i\).

Step 1: Express Square Roots with Negative Numbers in Terms of i

To handle the expression, let's first rewrite the square roots of negative numbers using the imaginary unit, where \(i = \sqrt{-1}\). The expression \(\sqrt{-24}\) can be rewritten as \(\sqrt{24} \cdot i\), because \(\sqrt{-1} = i\).

Step 2: Simplify the Expression

Now that we have square roots in terms of \(i\), let's simplify the fraction:\[ \frac{\sqrt{24} \cdot i}{\sqrt{6}} \]

Step 3: Simplify the Root Expression

Simplify the expression under the square root by treating \(\sqrt{24}\) and \(\sqrt{6}\):\[ \frac{\sqrt{24}}{\sqrt{6}} = \sqrt{\frac{24}{6}} = \sqrt{4} = 2 \]

Step 4: Combine and Simplify Further

Now that we have simplified the root expression, complete the simplification:\[ 2 \cdot i = 2i \]

Step 5: Final Simplification Step

The expression \(\frac{\sqrt{-24}}{\sqrt{6}}\), when simplified using \(i\), results in:\[ 2i \]

Key Concepts

Complex Numbers

Complex numbers are fascinating elements of mathematics that extend the idea of numbers beyond the real number line. A complex number consists of two parts: a real part and an imaginary part. The general form of a complex number is written as

• \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
• The letter \(i\) represents the imaginary unit defined as \(\sqrt{-1}\).

The imaginary unit \(i\) introduces the concept of numbers that are not on the traditional number line, making it possible to solve equations like \(x^2 + 1 = 0\) where no real solutions exist. Therefore, complex numbers are essential in many fields such as engineering, physics, and applied mathematics.
When handling complex numbers, it's important to remember that when multiplying or dividing these numbers, they behave like algebraic terms but with special rules for the imaginary unit:

• \(i^2 = -1\)

Simplifying Radicals

Simplifying radicals is about breaking down expressions under a square root so they are easier to understand and work with. Radicals often appear in problems where specific values need to be extracted under certain mathematical operations, just like in the original exercise. To simplify radicals:

• Divide the numbers under the square root as much as you can by finding perfect squares.
• For example, \(\sqrt{24}\) can be simplified as \(2\sqrt{6}\) because \(24\) is \(4 \times 6\) and \(\sqrt{4} = 2\).

In expressions including negative numbers under a square root, express them using the imaginary unit \(i\). For example, \(\sqrt{-24}\) becomes \(\sqrt{24} \cdot i\). This is a crucial step in simplifying complex expressions.
Radicals in the denominator can be managed using the property \(\sqrt{a}/\sqrt{b} = \sqrt{a/b}\), which simplifies examples like \(\frac{\sqrt{24}}{\sqrt{6}}\) into easier numbers to work with, such as \(\sqrt{4} = 2\). Always aim to express radicals in their simplest, most comprehensible form.

Algebraic Expressions

Algebraic expressions play a central role in various mathematical calculations and transformations, making understanding them essential. These expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Simplifying algebraic expressions involves combining like terms and using operations to make the expression more straightforward.
In the context of complex numbers and radicals, you often deal with expressions that involve both real and imaginary parts. For example, simplifying \(\frac{\sqrt{-24}}{\sqrt{6}}\) requires both understanding radicals and the imaginary unit simultaneously.
Approach each step carefully:

• Express each part in terms of simpler radicals or imaginary numbers, as discussed before.
• Combine the simplified parts using algebraic rules to achieve the neatest form possible.

The key to mastery is practice—frequently work through exercises, ensuring each operation is grounded in these principles. The comprehension of algebraic expressions is foundational for tackling more complex mathematical concepts.