Problem 44 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{12}{147}} $$

Step-by-Step Solution

Answer

\(\sqrt{\frac{12}{147}} = \frac{2}{7}\)

Step 1: Simplify the fraction

First, simplify the fraction by finding the greatest common divisor (GCD) of 12 and 147, which is 3. Divide both the numerator and the denominator by the GCD: \(\frac{12}{147} = \frac{4}{49}\).

Step 2: Simplify the root

Square root simplifies the expression even further. The square root of fraction is equal to square root of numerator divided by square root of denominator. \(\sqrt{\frac{4}{49}} = \frac{\sqrt{4}}{\sqrt{49}}\).

Step 3: Calculate the roots

Calculate the individual roots. \(\sqrt{4}\) is 2 and \(\sqrt{49}\) is 7. Therefore: \(\frac{\sqrt{4}}{\sqrt{49}} = \frac{2}{7}\)

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor or highest common factor, is a critical concept when it comes to simplifying mathematical expressions, especially fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

For instance, to simplify the expression \( \sqrt{\frac{12}{147}} \), we first need to find the GCD of 12 and 147. By listing the factors of both:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 147: 1, 3, 7, 21, 49, 147

The largest number appearing in both lists is 3. Therefore, 3 is the GCD of 12 and 147.

By dividing both the numerator and the denominator of the fraction by the GCD, we simplify the fraction to its lowest terms, \(\frac{4}{49}\) in this case. Understanding and finding the GCD is essential because it ensures that the fraction is in its simplest form before proceeding to further operations such as finding its square root.

Simplifying the Square Root of a Fraction

Simplifying the square root of a fraction may seem daunting, but it's quite straightforward when broken down into steps. The key is to remember that the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

Let's look at \(\sqrt{\frac{4}{49}}\) as an example. To simplify this, we take the square root of the numerator and the denominator separately. Hence, we calculate \(\sqrt{4}\) and \(\sqrt{49}\), which are 2 and 7, respectively. This gives us \(\frac{2}{7}\) after simplification.

It's crucial to simplify the fraction before taking the square roots to make the calculation easier and prevent error with larger numbers. Simplifying the initial fraction helps in revealing perfect squares in the numerator and the denominator, leading to whole numbers as results.

The Process of Simplifying Fractions

Simplifying fractions is a fundamental skill in math that helps make numbers more manageable and equations cleaner. To simplify a fraction, you want to reduce it to its simplest form, where the numerator and the denominator are as small as possible and have no common factors other than 1.

In our example, we started with the fraction \(\frac{12}{147}\) and found its GCD to be 3. By dividing both the top and bottom by this number, we get to a simpler equivalent fraction, \(\frac{4}{49}\). In general, simplifying a fraction involves:

• Finding the GCD of the numerator and denominator.
• Dividing both the numerator and denominator by the GCD.
• Checking the result to ensure it cannot be reduced any further.

When fractions are simplified, subsequent arithmetic operations become much easier, which is particularly useful when tackling more complex algebraic expressions.