Problem 11 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{8}{\sqrt{24}}\)

Step-by-Step Solution

Answer

\( \frac{8}{\sqrt{24}} = \frac{2\sqrt{6}}{3} \)

Step 1: Identify the Expression to Rationalize

We have the expression \( \frac{8}{\sqrt{24}} \). Our goal is to eliminate the square root in the denominator through rationalization.

Step 2: Simplify the Square Root in the Denominator

First, we simplify \( \sqrt{24} \). Notice that \( 24 = 4 \times 6 \), thus \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \). Substitute back to get \( \frac{8}{2\sqrt{6}} \).

Step 3: Reduce the Fraction

Simplify \( \frac{8}{2\sqrt{6}} \) by dividing the numerator and the coefficient in the denominator by 2. This gives \( \frac{4}{\sqrt{6}} \).

Step 4: Rationalize the Denominator

Multiply both numerator and denominator by \( \sqrt{6} \) to eliminate the square root in the denominator. This results in \( \frac{4 \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}} = \frac{4\sqrt{6}}{6} \).

Step 5: Simplify the Resulting Fraction

Finally, simplify \( \frac{4\sqrt{6}}{6} \) by dividing both the coefficient 4 in the numerator and 6 in the denominator by 2. This gives \( \frac{2\sqrt{6}}{3} \), which is the simplest form.

Key Concepts

Simplifying Radicals

Radicals, often involving square roots, appear frequently in various mathematical expressions. Simplifying these expressions helps in solving equations or rationalizing denominators. A radical is considered simplified when there are no perfect square factors remaining under the radical sign.

To simplify a radical like \( \sqrt{24} \), look for the highest perfect square that divides the number inside the radical. Here, 24 can be broken down into 4 times 6 (i.e., \( 24 = 4 \times 6 \)), and since 4 is a perfect square, it can be taken out of the square root as 2. This changes \( \sqrt{24} \) into \( 2\sqrt{6} \).

The process of breaking the number into its factor pairs and identifying perfect squares assists in reducing the complexity of the expression. While simplifying a radical, always ensure that the number under the square root is as small as possible and consists of no perfect square factors. This simplification is a crucial step before advancing to other operations like rationalizing the denominator.

Fractions

Understanding fractions is essential when working with more complex mathematical problems. A fraction consists of two parts: a numerator and a denominator. It represents a division of the numerator by the denominator.

In the context of rationalizing denominators, we often deal with fractions that have radicals in the denominator. The simplest form of a fraction is attained when both the numerator and the denominator cannot be divided any further by a common divisor except 1.

• For example, consider the fraction \( \frac{8}{\sqrt{24}} \).
• After simplification of the radical, it becomes \( \frac{8}{2\sqrt{6}} \), which can be further simplified by dividing both the numerator and the denominator by 2, giving \( \frac{4}{\sqrt{6}} \).

This simplification step is necessary before the process of rationalizing, as it makes the numbers more manageable and reduces potential calculation errors.

Square Roots

Square roots are the most common type of radicals and are fundamental in mathematics. The square root of a number \( x \) is another number y such that \( y^2 = x \). That is, \( y \) is a value that, when multiplied by itself, yields \( x \).

In expressions where square roots are present, particularly in denominators, the goal is often to remove them for ease of calculation. This removal, known as rationalizing the denominator, involves the use of the square root properties.

• In the provided exercise, to rationalize \( \frac{4}{\sqrt{6}} \), multiply both the numerator and denominator by \( \sqrt{6} \).
• This operation results in \( \frac{4\sqrt{6}}{6} \), eliminating the square root from the denominator.

This method works because \( \sqrt{6} \times \sqrt{6} = 6 \), thus converting a radical into an integer and simplifying the fraction. Understanding square roots and their properties is key to successfully manipulating such expressions in algebra.