Problem 14 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{\sqrt{24}}{2 \sqrt{6}}\)

Step-by-Step Solution

Answer

The simplified form is 1.

Step 1: Simplify the Square Root in the Numerator

Start by simplifying the square root of 24 in the numerator. The number 24 can be broken down into its prime factors: 24 = 2^3 * 3. Thus, \( \sqrt{24} = \sqrt{2^3 \times 3} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \sqrt{6} \). Rewrite the fraction: \( \frac{\sqrt{24}}{2 \sqrt{6}} = \frac{2 \sqrt{6}}{2 \sqrt{6}} \).

Step 2: Simplify the Fraction

Next, simplify the fraction by dividing both the numerator and denominator by \( 2 \sqrt{6} \). The fraction simplifies as follows: \( \frac{2 \sqrt{6}}{2 \sqrt{6}} = 1 \).

Step 3: Write in Simplest Form

The fraction has been fully simplified to its simplest form. The rationalized expression is now just the number 1.

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots is essential when working with expressions that involve them. Simplifying a square root entails breaking down the number inside the root into its prime factors. Once this breakdown is achieved, we aim to find pairs of factors that can be taken out of the square root.For instance, consider the square root of 24. To simplify:

• Break down 24 into its prime factors: 24 equals 2 multiplied by 2, multiplied by 2, and then by 3. So, it can be written as \( 24 = 2^3 \times 3 \).
• Look for pairs among the prime factors. Here, the pair is \( 2^2 \), which is 4. The square root of 4 is 2.
• The square root of 24 can be rewritten as \( \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \), which equals \( 2\sqrt{6} \).

Hence, \( \sqrt{24} \) is simplified to \( 2 \sqrt{6} \). This step is crucial before moving on to fraction simplification tasks.

Fraction Simplification

Once the square roots in a fraction are simplified, the next step is to simplify the fraction itself. Fraction simplification involves reducing the numerator and the denominator to their simplest form. This process generally includes dividing both by their greatest common factor.In our example, after simplifying the square root, the fraction \( \frac{2 \sqrt{6}}{2 \sqrt{6}} \) becomes apparent. Notice that the numerator and the denominator are identical:

• As a result, divide both the numerator and the denominator by \( 2 \sqrt{6} \).
• This division yields 1 since any non-zero number divided by itself is 1.

This process is vital as it reveals the simplest form of the fraction, removing any unnecessary complexity and preparing it for final presentation.

Prime Factorization

Prime factorization is a foundational math skill that is especially useful in simplifying square roots and fractions. It involves breaking down a number into a product of prime numbers. These prime numbers are the building blocks of the number and are crucial for deeper mathematical manipulations.Let's take the number 24 as an example:

• Begin by dividing 24 by the smallest possible prime number, which is 2. Since 24 is even, it works. Continuing this process: 24 divided by 2 gives 12.
• Divide 12 by 2 again to get 6, and then divide 6 by 2 to get 3.
• At 3, which is a prime number, we stop. Therefore, the prime factorization of 24 is \(2^3 \times 3\).

Identifying these prime factors can make simplifying expressions more manageable, as it helps in recognizing and grouping terms for simplification, making this an indispensable tool in algebra and beyond.