Problem 15 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{9}{36}}\)

Step-by-Step Solution

Answer

The answer is \(\frac{1}{2}\).

Step 1: Simplify the Fraction Inside the Square Root

Start by simplifying the fraction inside the square root: \(\frac{9}{36}\). Both the numerator and the denominator have a common factor of 9. Divide the numerator by 9 to get 1, and the denominator by 9 to get 4. This simplifies the fraction to \(\frac{1}{4}\).

Step 2: Apply Square Root to the Simplified Fraction

Apply the square root to the simplified fraction \(\frac{1}{4}\). The square root of a fraction can be taken as the square root of the numerator and the square root of the denominator separately. Thus, \(\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}\).

Step 3: Calculate the Square Roots

Calculate the square roots: \(\sqrt{1} = 1\) and \(\sqrt{4} = 2\). This gives \(\frac{1}{2}\) as the result of \(\frac{\sqrt{1}}{\sqrt{4}}\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their most basic form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF). In the context of the problem, the fraction inside the square root is \( \frac{9}{36} \).

• First, identify a number that evenly divides both the numerator and the denominator. In this case, the number is 9.
• Next, divide both the top number (9) and the bottom number (36) by this common factor. This process transforms the fraction into \( \frac{1}{4} \), a much simpler expression.

By simplifying fractions, we make calculations more manageable and understand the relationships between numbers more clearly.

Common Factors

Common factors are numbers that divide exactly into two or more bigger numbers. For finding the common factor of a fraction, look at the numerator and the denominator to see if there's a number they both can be divided by evenly.

• In \( \frac{9}{36} \), the number 9 is a common factor because it can divide both 9 and 36 without leaving a remainder.
• If both the numerator and the denominator have more than one common factor, choose the greatest one to simplify as much as possible.

Identifying common factors is an essential first step in simplifying fractions, which in turn can simplify other mathematical operations like taking square roots.

Square Roots of Fractions

Finding the square root of a fraction may seem complex, but it can be easier when broken down into straightforward steps. When you have a fraction under a square root, you can simplify the process by taking the square root of both the numerator and the denominator separately.

• For the fraction \( \frac{1}{4} \), you find the square root of 1 (the numerator), which is 1.
• Then, find the square root of 4 (the denominator), which is 2.

This gives us \( \frac{1}{2} \) as the simplified result. Basically, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Understanding this method helps in solving any similar problems involving square roots of fractions.