Problem 24 - Fundamentals of Mathematics

Question

Find the value of each of the following. $$ \sqrt{\frac{1}{4}} $$

Step-by-Step Solution

Answer

The value is \( \frac{1}{2} \).

Step 1: Understanding the Square Root

The problem asks us to find the square root of the fraction \( \frac{1}{4} \). The square root of a number \( x \) is another number \( y \) such that \( y^2 = x \). Our task is to find \( y \) such that \( y^2 = \frac{1}{4} \).

Step 2: Simplify the Fraction

Recognize that the fraction \( \frac{1}{4} \) can be expressed as \( \frac{1}{2^2} \). This understanding will help simplify finding the square root.

Step 3: Calculate the Square Root of the Fraction

The square root of a fraction \( \frac{a}{b} \) is \( \frac{\sqrt{a}}{\sqrt{b}} \). Applying this, we have \( \sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} \).

Step 4: Calculate the Square Roots of the Numerator and Denominator

The square root of \( 1 \) is \( 1 \), and the square root of \( 4 \) is \( 2 \). Hence, \( \sqrt{\frac{1}{4}} = \frac{1}{2} \).

Step 5: Verify the Answer

To verify, square \( \frac{1}{2} \) and see if it equals \( \frac{1}{4} \). Calculating, \( \left( \frac{1}{2} \right)^2 = \frac{1^2}{2^2} = \frac{1}{4} \), which confirms our answer is correct.

Key Concepts

Understanding Fractions

Fractions are a way to express numbers that are not whole. They consist of two parts: a numerator and a denominator.

• The numerator is the top part of the fraction, indicating the number of parts you have.
• The denominator is the bottom part, showing how many parts make up a whole.

In our exercise, the fraction is \( \frac{1}{4} \). Here, \( 1 \) represents the numerator, and \( 4 \) is the denominator. This means we have one out of four equal parts of a whole. Fractions are flexible and can represent various things such as parts of a pizza, segments of a whole, or even results of division.

The Process of Simplification

Simplification of a fraction is the process of converting it to its simplest form. This means making the numerator and the denominator as small as possible while keeping the value of the fraction the same.For instance, if we have the fraction \( \frac{2}{8} \), both numerator and denominator can be divided by 2, simplifying it to \( \frac{1}{4} \). In our original exercise, the fraction is already in its simplest form.Simplification is crucial in ensuring computations are easier and results are more readable. By recognizing patterns, like how \( \frac{1}{4} \) can relate to other fractions, we can tackle mathematical problems more efficiently.

Decoding the Numerator

The numerator of a fraction plays an important role in the value the fraction represents. It's essentially the "counter" of how many parts we are considering.In the fraction \( \frac{1}{4} \), the numerator is \( 1 \). It is what we seek the square root of when simplifying square roots of fractions as per our previous step-by-step solution.The square root of the numerator, \( \sqrt{1} \), is simply \( 1 \), because 1 times 1 stays the same. Understanding how to work with numerators helps in calculating precise results in problems involving operations on fractions.

Explaining the Denominator

The denominator is the bottom part of a fraction, and it shows into how many parts the whole is divided.In our fraction \( \frac{1}{4} \), the denominator is \( 4 \). It denotes that the whole has been divided into four equal parts. When we calculate the square root of a fraction, we also need to consider the square root of the denominator.For \( \sqrt{4} \), the result is \( 2 \), since \( 2 \times 2 = 4 \).Understanding denominators is essential as they set the fractional context for any calculation. They make sure we recognize what "whole" we are dealing with and help us make accurate mathematical interpretations of fractions.