Problem 54 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{1}{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: $\sqrt{\frac{1}{4}}$ Answer: $\frac{1}{2}$

Step 1: Recall the properties of radicals

Recall the following property of radicals: $$ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$ We will use this property to separate the square root of the fraction into the square roots of the numerator and the denominator.

Step 2: Apply the property of radicals to the given expression

Applying the property of radicals we mentioned earlier to our radical expression: $$ \sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} $$

Step 3: Simplify the square roots in the numerator and denominator

Now, we just need to take the square root of the numerator and the denominator: $$ \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2} $$ So, the simplified radical expression is \(\frac{1}{2}\).

Key Concepts

Properties of Radicals

Understanding the properties of radicals is essential when simplifying radical expressions, especially when working with fractions. One key property to remember is how to handle the square root of a division:

• When you see a square root over a fraction like \( \sqrt{\frac{a}{b}} \), you can simplify it by separating it into two separate roots: \( \frac{\sqrt{a}}{\sqrt{b}} \).

This means you can deal with the numerator and the denominator independently, which makes calculations easier.
It’s a powerful tool that helps keep your work organized and clear. Remember that this property not only works with square roots but also with other types of roots, such as cube roots. However, the symbols inside must always be non-negative for the property to hold true.

Square Roots

The concept of square roots is straightforward but crucial in many areas of mathematics. A square root asks what number, when multiplied by itself, gives the original number. For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

• This principle applies to simplifying radicals by breaking them down into their basic root form.

Recognizing perfect squares—numbers like 1, 4, 9, 16, and so on—can make the process of finding square roots much simpler.
In our exercise, knowing that \( \sqrt{1} = 1 \) and \( \sqrt{4} = 2 \) is critical in simplifying the expression to \( \frac{1}{2} \). Being familiar with these helps you calculate square roots quickly and efficiently without any additional steps.

Fractional Expressions

Fractional expressions under a radical can initially seem daunting, but breaking them down into their components often simplifies them significantly. The fundamental idea is converting the expression to separate the numerator and denominator's square roots, as our property of radicals suggests.

• For example, through this technique, \( \sqrt{\frac{1}{4}} \) becomes \( \frac{\sqrt{1}}{\sqrt{4}} \).

Once separated, each part of the fraction can be simplified individually.
Remember that simplifying fractional expressions and understanding the division of square roots go hand-in-hand. The goal is to reach an expression where the radical is simplified entirely, making further operations or explanations much clearer. This approach not only simplifies mathematical expressions but also aids in understanding and solving real-world problems involving fractions and radicals.