Problem 16 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{144}{36}}\)

Step-by-Step Solution

Answer

The result is 2.

Step 1: Identify the Expression

The exercise requires you to evaluate the expression \( \sqrt{\frac{144}{36}} \). This means we need to simplify the fraction inside the square root first.

Step 2: Simplify the Fraction

Start by simplifying the fraction inside the square root: \( \frac{144}{36} \). Divide both the numerator and the denominator by their greatest common divisor. Here, \(144 \div 36 = 4\). So, \(\frac{144}{36} = 4\).

Step 3: Apply the Square Root

Now that the fraction is simplified to 4, apply the square root: \( \sqrt{4} \). The square root of 4 is 2 because \(2 \times 2 = 4\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key mathematical skill that helps make fractions easier to understand or work with. When we simplify a fraction, we express it in the simplest form by reducing the numerator (the top part) and the denominator (the bottom part) to the smallest possible whole numbers while keeping the same value.
To simplify a fraction, we need to:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

In our example, before simplifying, we had the fraction \( \frac{144}{36} \). By using simplification, we reduced this fraction to 4 since \( 144 \div 36 \) is equal to 4. This step is crucial before proceeding with evaluating the square root.

Greatest Common Divisor

The greatest common divisor (GCD) plays an important role in simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD allows us to simplify fractions correctly.
To find the GCD:

• List the factors of the numerator and the denominator.
• Identify the largest common factor shared by both.

In the fraction \( \frac{144}{36} \), both 144 and 36 are divisible by 36. Therefore, the GCD is 36. By dividing the numerator and the denominator by 36, the fraction becomes simplified to \( \frac{144}{36} = 4 \). Understanding the GCD helps us simplify our mathematical expressions correctly.

Mathematical Expressions

Mathematical expressions include numbers, variables, and operations that combine together to represent a value or a relation. In the context of the original exercise, our expression was \( \sqrt{\frac{144}{36}} \). This expression involves a fraction under a square root.
Evaluating such expressions requires breaking them down into manageable steps:

• Start by simplifying any fractions, as simplification makes subsequent calculations easier.
• Apply the appropriate mathematical operations, like evaluating the square root once the fraction is simplified.

Once the expression is simplified inside the square root, you can easily evaluate it. For example, after simplifying \( \frac{144}{36} \) to 4, the expression \( \sqrt{4} \) becomes much simpler to evaluate, resulting in 2. This structured approach helps in working through complex expressions efficiently.