Problem 18 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{336}}{\sqrt{21}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression \(\frac{\sqrt{336}}{\sqrt{21}}\) is \(2\sqrt{2}\).

Step 1: Simplify the square root of the numerator

First, we need to find the prime factorization of 336. The prime numbers that multiply to get 336 are 2, 2, 2, 3, and 7. Mathematically, we can write this as: $$ 336 = 2^3 \cdot 3^1 \cdot 7^1 $$ Now we can simplify the square root of 336: $$ \sqrt{336} = \sqrt{2^3 \cdot 3^1 \cdot 7^1} = 2\sqrt{2 \cdot 3 \cdot 7} $$

Step 2: Simplify the square root of the denominator

To find the prime factorization of 21, we should find the prime numbers that multiply to get 21. In this case, 3 and 7 are the prime numbers. Mathematically, we can write this as: $$ 21 = 3^1 \cdot 7^1 $$ Now we can simplify the square root of 21: $$ \sqrt{21} = \sqrt{3^1 \cdot 7^1} = \sqrt{3 \cdot 7} $$

Step 3: Divide the simplified numerator by the simplified denominator

Now we can divide the simplified expression in the numerator by the simplified expression in the denominator: $$ \frac{\sqrt{336}}{\sqrt{21}} = \frac{2\sqrt{2 \cdot 3 \cdot 7}}{\sqrt{3 \cdot 7}} $$ Here, we can cancel out the common terms in the numerator and the denominator, which are \(\sqrt{3}\) and \(\sqrt{7}\): $$ \frac{2\sqrt{2 \cdot 3 \cdot 7}}{\sqrt{3 \cdot 7}} = 2\sqrt{2} $$

Step 4: Write the final answer

The simplified expression of the given problem is: $$ \frac{\sqrt{336}}{\sqrt{21}} = 2\sqrt{2} $$

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Understanding prime factorization is crucial in simplifying radicals, as it helps break down a number into its simplest form.

For example, consider the number 336. To find its prime factorization, we start by dividing by the smallest prime number, which is 2. We divide 336 by 2 to get 168, again by 2 to get 84, and by 2 once more to get 42. Then, we divide by 3 to get 14, and finally by 7 to get 1. Thus, the prime factorization of 336 is expressed as:

• 336 = 2 × 2 × 2 × 3 × 7

By repeating this method for 21, we find:

• 21 = 3 × 7

This step is the foundation to simplifying square roots. Once a number is broken into prime factors, pairing identical factors can simplify the square root operation.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. The square root symbol is \(\sqrt{}\). Simplifying square roots involves expressing them in their simplest form using prime factorization.

For instance, take the square root of 336. Using its prime factorization \(336 = 2^3 \cdot 3^1 \cdot 7^1\), we look for pairs of prime factors because each pair inside the root can be simplified to a single number outside the root. In this case, \(2^3\) contains one pair of 2s and one leftover 2:

• \(\sqrt{336} = \sqrt{2^3 \cdot 3^1 \cdot 7^1} = 2\sqrt{2 \cdot 3 \cdot 7}\)

Similarly, to simplify \(\sqrt{21}\), we use its prime factorization \(21 = 3^1 \cdot 7^1\). No pairs exist here, so it remains:

• \(\sqrt{21} = \sqrt{3 \cdot 7}\)

Recognizing squares within factors is key to simplifying radicals efficiently.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Simplification often involves reducing the expression to its simplest form, similar to numerical fractions.

In the context of our exercise, simplifying an algebraic fraction means breaking the fraction into more manageable parts using factors. In our example \(\frac{\sqrt{336}}{\sqrt{21}}\), both the numerator and the denominator are square roots. After simplifying the square roots using prime factorization, we have:

• Numerator: \(2\sqrt{2 \cdot 3 \cdot 7}\)
• Denominator: \(\sqrt{3 \cdot 7}\)

By canceling the common terms \(\sqrt{3}\) and \(\sqrt{7}\) from the numerator and denominator, the expression simplifies to:

• Final result: \(2\sqrt{2}\)

This process highlights the importance of finding shared factors in numerator and denominator, which allows the simplification of complex expressions. Understanding how to manipulate algebraic fractions is essential for solving problems efficiently in algebra.