Problem 54 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt{10}}{\sqrt{20}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{2}}{2}\) is the simplest radical form.

Step 1: Simplify the Radical Fraction

The given expression is \( \frac{\sqrt{10}}{\sqrt{20}} \). According to the property of radicals \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), we can rewrite this as \( \sqrt{\frac{10}{20}} \).

Step 2: Simplify the Fraction Inside the Radical

Simplify the fraction inside the radical: \( \frac{10}{20} = \frac{1}{2} \). Now our expression is \( \sqrt{\frac{1}{2}} \).

Step 3: Rationalize the Denominator

Multiply the numerator and the denominator by \( \sqrt{2} \) to remove the square root from the denominator: \( \sqrt{\frac{1}{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \).

Step 4: Verify the Simplified Form

The expression \( \frac{\sqrt{2}}{2} \) is in its simplest radical form. We cannot simplify it further as there are no perfect squares in the numerator that we can reduce. The final answer is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Radical Simplification

Radical simplification involves changing a radical expression into its simplest form. In our example, we started with the expression \( \frac{\sqrt{10}}{\sqrt{20}} \). A key property of radicals allows us to rewrite this as a single square root of a fraction: \( \sqrt{\frac{10}{20}} \). This transformation simplifies the process by focusing on simplifying the fraction first.

Simplifying radicals follows some basic steps:

• Combine or separate radicals using basic arithmetic principles especially when dealing with fractions or products.
• Look for factors that are perfect squares, which makes simplifying easier.

For example, when we had \( \sqrt{\frac{10}{20}} \), we could simplify the fraction \( \frac{10}{20} = \frac{1}{2} \), leading us to a simpler expression \( \sqrt{\frac{1}{2}} \).

This simplification step reduces complications and prepares expressions for further operations like rationalizing denominators.

Rationalizing the Denominator

Rationalizing the denominator is an essential step when simplifying radical expressions. The goal is to eliminate radicals from the denominator, making calculations more straightforward. In the example \( \sqrt{\frac{1}{2}} \), the denominator is still in a radical form. So, we multiply both the numerator and the denominator by \( \sqrt{2} \). This is because multiplying by \( \sqrt{2} \) will convert the denominator into a rational number.

Here's a brief breakdown of the process:

• Multiply both the numerator and denominator by a radical that will remove the square root from the denominator.
• In this case, multiplying by \( \sqrt{2} \) gives \( \sqrt{\frac{1}{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \).

The result \( \frac{\sqrt{2}}{2} \) is cleaner and easier to work with in further mathematical operations. Rationalizing helps maintain the simplicity and clarity needed for effective communication and calculation in math.

Properties of Radicals

Understanding properties of radicals is crucial for simplifying expressions effectively. Radicals, particularly square roots, have certain rules:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \): This property helps in combining or separating square roots as needed.
• \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \): Used to simplify fractions under a common radical, as seen in our example.

These properties are foundational tools that enable us to manipulate and simplify expressions.Applying the properties correctly can transform a complex expression into something more manageable and better suited for further calculation or interpretation, like transforming \( \frac{\sqrt{10}}{\sqrt{20}} \) into \( \frac{\sqrt{2}}{2} \). Understanding these properties makes working with radicals both efficient and intuitive, enhancing overall mathematical fluency.