Problem 14 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{16}{64}}\)

Step-by-Step Solution

Answer

\( \sqrt{\frac{16}{64}} = \frac{1}{2} \).

Step 1: Simplify the Fraction Under the Square Root

The expression given is \( \sqrt{\frac{16}{64}} \). Start by simplifying the fraction inside the square root. To do this, divide both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 16 and 64 is 16. Thus, \( \frac{16}{64} = \frac{16 \div 16}{64 \div 16} = \frac{1}{4} \).

Step 2: Apply the Square Root to the Simplified Fraction

Now that the fraction inside the square root is simplified to \( \frac{1}{4} \), we can apply the square root to both the numerator and the denominator separately. This gives \( \sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} \).

Step 3: Evaluate the Square Roots of the Numerator and Denominator

Calculate the square root of the numerator and the denominator: \( \sqrt{1} = 1 \) and \( \sqrt{4} = 2 \).

Step 4: Simplify the Result

With the square roots calculated, the expression becomes \( \frac{1}{2} \). Therefore, \( \sqrt{\frac{16}{64}} = \frac{1}{2} \).

Key Concepts

Simplifying Fractions

Fractions consist of a numerator and a denominator, each a part of a whole. Simplifying fractions helps make calculations easier by reducing the fraction to its simplest form.
For \( \frac{16}{64} \), this means finding a smaller equivalent fraction that has the same value. To simplify a fraction:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD to reduce the fraction.

Once simplified, \( \frac{16}{64} \) reduces to \( \frac{1}{4} \). This keeps the fraction's value the same but in a smaller, neater form.

Greatest Common Divisor

Understanding the greatest common divisor is essential for simplifying fractions. The GCD of two numbers is the largest number that divides both without a remainder.
Finding the GCD involves:

• Listing the factors of each number.
• Identifying the largest factor common to both lists.

In this specific exercise, the GCD of 16 and 64 is 16. This means 16 is the largest number that both can be divided by evenly, making it the key to simplifying \( \frac{16}{64} \) to \( \frac{1}{4} \).
This step is crucial for making the square root calculation straightforward.

Numerator and Denominator

Each fraction comprises two parts: the numerator and the denominator. The numerator is the top number, representing how many parts are being considered. The denominator is the bottom number, indicating the total number of equal parts.
When simplifying or evaluating a fraction, both numerator and denominator are important. They guide the process of simplifying with the GCD.
For example, in the fraction \( \frac{1}{4} \), 1 is the numerator, and 4 is the denominator.

• The numerator is used in calculations such as determining the square root.
• The denominator dictates how the fraction's parts are divided.

Simplifying fractions and finding square roots necessitates working with both components of a fraction to arrive at an accurate result.

Mathematical Evaluation

The process of mathematical evaluation involves performing operations to obtain values. In this exercise, evaluation guides us from problem to solution.
Start by simplifying the fraction, \( \frac{16}{64} = \frac{1}{4} \), making evaluation easier by using manageable numbers.

• First, apply the square root to the numerator and denominator separately.\( \sqrt{1} = 1 \)
• Next, find the square root of \( 4 \). This is \( \sqrt{4} = 2 \).

This allows us to evaluate the expression step by step. Finalizing this evaluation: \( \sqrt{\frac{1}{4}} = \frac{1}{2} \). Thus, the simple division at the end ensures complete and correct evaluation.