Problem 25 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{4}{225}}$$

Step-by-Step Solution

Answer

\(\frac{2}{15}\)

Step 1: Simplify the Square Root of a Fraction

To simplify the square root of a fraction \( \sqrt{\frac{4}{225}} \), apply the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This gives us \( \frac{\sqrt{4}}{\sqrt{225}} \).

Step 2: Calculate the Square Roots of the Numerator and Denominator

The square root of 4 is 2, because \( 2^2 = 4 \). Similarly, the square root of 225 is 15, because \( 15^2 = 225 \). Thus, \( \frac{\sqrt{4}}{\sqrt{225}} = \frac{2}{15} \).

Key Concepts

Square Root Property

When dealing with radical expressions, especially those involving fractions, the square root property becomes very useful. The square root property for a fraction stipulates that the square root of a fraction \( \sqrt{\frac{a}{b}} \) is equal to the fraction composed of the square roots of its numerator and denominator. This can be expressed mathematically as:

• \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

This property is key when simplifying complex expressions. It allows each part of the fraction to be addressed separately, making the entire expression more manageable. Applying the square root property to an initial expression splits the problem into two easier tasks: finding the square roots of simple numbers, which is typically straightforward. Remember this property as a tool for breaking down problems and simplifying your calculations.

Numerator and Denominator

Fractions comprise a numerator and a denominator. The numerator is the upper part of the fraction, representing the number of parts we have. In contrast, the denominator is the lower part, symbolizing the total number of equal parts into which the whole is divided. Understanding this structure is important when simplifying expressions like \( \sqrt{\frac{4}{225}} \), as each component must be tackled individually.

• For \( \sqrt{\frac{4}{225}} \), the numerator is 4
• The denominator is 225

By dealing with the numerator and denominator separately, you ensure accuracy while simplifying. After applying the square root property, you simplify by calculating the square roots of these individual components before recombining them into a final, simplified fraction.

Square Root Calculation

Calculating square roots is essential for fully applying the square root property. Square roots ask, "What number, when multiplied by itself, produces this number?" For instance, the square root of 4 is 2 because \( 2 \times 2 = 4 \). Similarly, the square root of 225 is 15 because \( 15 \times 15 = 225 \).

• \( \sqrt{4} = 2 \)
• \( \sqrt{225} = 15 \)

In the given problem, you calculate these square roots separately after applying the property to the fraction and then recombine these results to achieve the final simplified expression, \( \frac{2}{15} \). These steps ensure that you have simplified the expression correctly and completely.