Problem 39 - A Graphical Approach to College Algebra

Question

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-24}}{\sqrt{8}}$$

Step-by-Step Solution

Answer

The simplified expression is \(i\sqrt{3}\).

Step 1: Simplify the Radical in the Numerator

Notice that we have a negative number under the radical in the numerator, which indicates the presence of an imaginary number. Therefore, \( \sqrt{-24} = i \sqrt{24} \). Simplifying \( \sqrt{24} \), we can rewrite it as \( \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \). Thus, the expression becomes \( i \cdot 2\sqrt{6} \).

Step 2: Simplify the Radical in the Denominator

The denominator \( \sqrt{8} \) can be simplified by recognizing that \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \).

Step 3: Combine the Fraction

Substitute the simplified radical expressions back into the original fraction: \( \frac{i \cdot 2\sqrt{6}}{2\sqrt{2}} \).

Step 4: Simplify the Rational Expression

Cancel the common factor of 2 in both the numerator and denominator, giving \( \frac{i\sqrt{6}}{\sqrt{2}} \).

Step 5: Further Simplify the Expression

To simplify \( \frac{i\sqrt{6}}{\sqrt{2}} \), multiply the numerator and the denominator by \( \sqrt{2} \): \( \frac{i\sqrt{6} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{i\sqrt{12}}{2} \).

Step 6: Simplify the Square Root in the Numerator

Recognize that \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \). So our expression becomes \( \frac{i \cdot 2\sqrt{3}}{2} \).

Step 7: Cancel Common Factors

Cancel the factor of 2 in the numerator and denominator, resulting in \( i\sqrt{3} \).

Key Concepts

Simplifying Radicals

When dealing with radicals, one of the most intriguing parts is simplifying them, particularly when they contain negative numbers under the root. To understand the idea better, consider the expression \( \sqrt{-24} \). Since we have a negative number under the square root, it introduces the concept of imaginary numbers. The imaginary unit "i" is defined such that \( i^2 = -1 \), which helps us express \( \sqrt{-24} \) as \( i \sqrt{24} \).
Now, to further simplify \( \sqrt{24} \), we look for perfect square factors. Here, \( 24 \) can be broken down into \( 4 \times 6 \), where 4 is a perfect square. Thus, \( \sqrt{24} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \).
This simplification process can be applied to all radicals. You search for the largest perfect square factor and rewrite the expression using that factor, making calculations and expressions less cumbersome and more elegant in subsequent steps.

Complex Numbers

Complex numbers are fascinating entities in mathematics, combining the real and the imaginary. Any number of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, is a complex number. In the exercise, we encounter complex numbers when simplifying radicals with negative numbers beneath the root.
The imaginary unit \( i \) is central to this, allowing us to work with expressions like \( \sqrt{-24} \). Once the imaginary part is isolated using \( i \), the remainder of the expression can be aligned with its real part.
In computations like division or multiplication, complex numbers follow a specific set of rules that maintain their structure. Understanding these numbers paves the way for delving into further mathematical concepts such as complex functions and transformations, which are vital in fields involving complex computations and signals.

Mathematical Expressions Simplification

Simplifying mathematical expressions often involves identifying common factors and performing operations such as multiplication or division efficiently. In our solution, dividing the expressions involves simplifying both the numerator and the denominator separately.
The core idea is to find any shared factors between the numerator and the denominator, which simplifies the expression by cancelling them out. In the exercise, after simplifying \( \sqrt{-24} \) to \( i2\sqrt{6} \) in the numerator and \( \sqrt{8} \) to \( 2\sqrt{2} \) in the denominator, we combined them into a single fraction: \( \frac{i \cdot 2\sqrt{6}}{2\sqrt{2}} \). Here, the common factor of 2 was cancelled.
Further simplification is achieved by multiplying the top and bottom by \( \sqrt{2} \), which makes the denominator a perfect square. Resolve any remaining radicals and factor elements such as powers of "i", to reach the simplest form. This approach reduces complexity and aids in clearer insight when working with intricate mathematical problems.