Problem 15 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{2}{5}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{\frac{2}{5}} \) is \( \frac{\sqrt{10}}{5} \)

Step 1: Break down the radical into two square roots

Radical of the fraction can be broken down into two radicals, i.e. square root of the numerator over square root of the denominator. This yields: \( \sqrt{\frac{2}{5}} = \frac{\sqrt{2}}{\sqrt{5}} \)

Step 2: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{5}\), giving: \( \frac{\sqrt{2}}{\sqrt{5}} = \frac{\sqrt{2} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{10}}{5} \)

Key Concepts

Rationalize the Denominator

When you encounter a fraction with a square root in the denominator, such as \( \frac{1}{\sqrt{2}} \), it's considered good mathematical form to remove the radical. This process is called 'rationalizing the denominator'. It’s not just about aesthetics; it allows for easier arithmetic operations and comparison of magnitudes. To rationalize the denominator, you need to eliminate the square root from the bottom of the fraction.

Let's take for instance our exercise \( \sqrt{\frac{2}{5}} \). Here, the denominator under the radical is 5. To rationalize, you multiply both the numerator and the denominator by the square root that’s in the denominator, \( \sqrt{5} \) in this case. This operation ensures that the denominator becomes a rational number, which in this example results in \( \frac{\sqrt{2} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{\sqrt{10}}{5} \). Now, you have a simplified expression with a rational denominator.

Radical Expressions

A radical expression is any mathematical expression containing a square root, cube root, or higher roots. They are critical in algebra and must be handled correctly to solve equations and simplify expressions. In a radical expression, the number under the root symbol is called the 'radicand'.

For instance, with the expression \( \sqrt{\frac{2}{5}} \), the radicands are 2 and 5 for the numerator and the denominator respectively. Dealing with a radical expression requires understanding its rules, such as the fact that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) and \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), which are utilized when breaking down or combining radicals.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression as simply as possible. This process often includes rationalizing denominators, as seen above, but it also entails breaking down the radical into its simplest radical form, combining like terms, and eliminating any perfect square factors from under the square root.

In our exercise, we started with the expression \( \sqrt{\frac{2}{5}} \) and simplified by splitting it into \( \frac{\sqrt{2}}{\sqrt{5}} \). However, our task is not complete until we address the square root in the denominator. By rationalizing and removing the radical from the denominator, we reach a more simplified form, \( \frac{\sqrt{10}}{5} \). As part of simplifying, we should also be on the lookout for simplifying the resultant radicals further if possible. In this case, since \( \sqrt{10} \) cannot be simplified any further, we have reached the simplest form.