Problem 16 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

2

Step 1: Simplify the Fraction Inside the Square Root

First, simplify the fraction \( \frac{144}{36} \). To do this, divide both the numerator and the denominator by 36. This gives \( \frac{144 \div 36}{36 \div 36} = \frac{4}{1} = 4 \).

Step 2: Find the Square Root of the Simplified Number

Now, take the square root of the simplified number, which is 4. We have \( \sqrt{4} = 2 \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common factor (GCF). This makes calculations easier and results more comprehensible.
To simplify a fraction, follow these steps:

• Identify the GCF of the numerator and the denominator. This is the largest number that can divide both the numerator and denominator without leaving a remainder.
• Divide both the numerator and the denominator by the GCF.
• Rewrite the fraction using the results from the division.

The simplified fraction should have no common factors between the numerator and the denominator other than 1. In the example given, the GCF of 144 and 36 is 36. Dividing both the numerator and the denominator by 36 results in the simplified fraction \(\frac{4}{1}\), which is equal to 4.

Numerator and Denominator

In a fraction, the numerator and the denominator are the two parts that make up the fraction itself. The numerator is the top number of the fraction, while the denominator is the bottom number.

• The numerator represents the number of parts we have or are considering.
• The denominator tells us into how many parts the whole is divided.

Understanding these roles is crucial when working with fractions in operations such as addition, subtraction, or in this case, simplifying.
In the fraction \(\frac{144}{36}\), 144 is the numerator and 36 is the denominator. Properly handling these values by dividing with the GCF allows us to simplify the fraction, as provided in the solution.

Division

Division is a fundamental arithmetic operation used to divide one number by another. In the context of simplifying fractions, division is used both to find the GCF and to apply it to the fraction itself.

• Division involves determining how many times the denominator can be perfectly fitted into the numerator.
• It is essential in breaking down fractions to their simplest form.

In the exercise, division plays a key role in simplifying \(\frac{144}{36}\). By dividing both the numerator and the denominator by 36, we found that \(\frac{144 \div 36}{36 \div 36} = \frac{4}{1}\). Performing division in this manner ensures the fraction is reduced effectively.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of mathematics and are essential for solving a variety of math problems, including simplifying fractions.

• Understanding each operation is crucial to knowing when and how to use them.
• Simplifying fractions often requires both multiplication and division, as seen in recognizing factors and applying the GCF.

Though we focused on division in this exercise, knowing how each operation can interact is vital. For example, simplifying \(\frac{144}{36}\) started with identifying multiplication factors, leading us to the division for simplification. This step is just one instance where basic operations come together to simplify complex mathematical processes.