Problem 25 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{\frac{4}{225}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{15} \).

Step 1: Simplify the Fraction Inside the Square Root

First, identify the fraction inside the square root, which is \( \frac{4}{225} \). The goal is to simplify this fraction before dealing with the square root. Since 4 and 225 are not obvious simplifications, confirm they have no common factors aside from 1. Hence, this fraction is already in its simplest form.

Step 2: Apply the Square Root to the Numerator and the Denominator

Since the square root of a fraction is the quotient of the square roots of the numerator and the denominator, separate \( \sqrt{\frac{4}{225}} \) into \( \frac{\sqrt{4}}{\sqrt{225}} \). This simplifies the expression into different parts.

Step 3: Calculate the Square Root of the Numerator

The numerator is 4. The square root of 4 is 2, since \(2 \times 2 = 4\). Therefore, \( \sqrt{4} = 2\).

Step 4: Calculate the Square Root of the Denominator

The denominator is 225. Notice that the square root of 225 is 15, because \(15 \times 15 = 225\). Thus, \( \sqrt{225} = 15 \).

Step 5: Combine the Results

Now, using the results from the previous steps, combine them to simplify the original expression. Therefore, \( \sqrt{\frac{4}{225}} = \frac{\sqrt{4}}{\sqrt{225}} = \frac{2}{15} \). The expression is now simplified.

Key Concepts

Square Roots

Square roots are fundamental in mathematics, providing a way to simplify expressions that involve squaring numbers. A square root, symbolized as \( \sqrt{} \), finds a number that, when multiplied by itself, results in the original number.

• For example, \( \sqrt{4} \) equals 2 because \( 2 \times 2 = 4 \).
• Similarly, \( \sqrt{225} \) equals 15 since \( 15 \times 15 = 225 \).

The square root faces, like our example \( \sqrt{\frac{4}{225}} \), can seem complex, but breaking them down by the numerator and the denominator makes this concept easier to handle. Always remember to simplify as much as possible by breaking down both the numerator and the denominator of the fraction before taking the square root, if it simplifies the process.

Fractions

Fractions represent a part of a whole and involve two numbers: a numerator (the top number) and a denominator (the bottom number). Fractions can be simplified by dividing both the numerator and the denominator by their greatest common factor.

• In our case, the fraction is \( \frac{4}{225} \).
• It is already in its simplest form because the only common factor is 1.

If a fraction is not in the simplest form, simplify it first before proceeding with any further operations, such as finding the square root. Having an unsimplified fraction can make other operations more complicated and error-prone.

Numerator and Denominator

Understanding the roles of numerators and denominators is crucial when working with fractions. The numerator represents how many parts you have, while the denominator tells you the size of those parts in relation to a whole.

• Here, in \( \frac{4}{225} \), 4 is the numerator, and 225 is the denominator.
• Purely focus on them separately when simplifying the square root of a fraction.

In the solution, we found the square root of the numerator and denominator separately—\( \sqrt{4} = 2 \) and \( \sqrt{225} = 15 \) respectively. By solving these individually, we ease the complexity and step towards the overall simplification.

Simplification Process

The simplification process involves expressing a mathematical expression in its most reduced form. This process is essential to achieving clearer, more straightforward results. Here’s how it applies to our example:1. **Identify the Fraction:** Recognize \( \frac{4}{225} \) is a fraction contained within a square root.2. **Apply the Square Root Separately:** Use this key technique to break \( \sqrt{\frac{4}{225}} \) into \( \frac{\sqrt{4}}{\sqrt{225}} \).3. **Calculate each Part:** Find \( \sqrt{4} \) and \( \sqrt{225} \)—calculated as 2 and 15 respectively.4. **Combine Results Efficiently:** This gives \( \frac{2}{15} \), a simplified version of the expression.Breaking down each step ensures you aren’t overwhelmed by the numbers and the mathematical operations involved. Ultimately, simplifying expressions reduces potential errors and provides a clear understanding of the final outcome.