Problem 45 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{2}{7}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{14}}{7}\) is the simplest radical form of \(\sqrt{\frac{2}{7}}\).

Step 1: Simplify the Fraction

Check if the fraction inside the radical can be simplified. Here, \(\frac{2}{7}\) is already in its simplest form, so we'll leave it as is.

Step 2: Separate the Radical

Use the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) to separate the radical into two parts: \(\sqrt{2}\) and \(\sqrt{7}\). This gives us \(\frac{\sqrt{2}}{\sqrt{7}}\).

Step 3: Rationalize the Denominator

To eliminate the square root from the denominator, multiply both the numerator and the denominator by \(\sqrt{7}\). This results in: \(\frac{\sqrt{2} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{\sqrt{14}}{7}\).

Step 4: Final Simplified Form

Ensure the fraction is simplified completely. Since \(\frac{\sqrt{14}}{7}\) cannot be simplified further, this is the simplest radical form of \(\sqrt{\frac{2}{7}}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is generally done to make the calculations easier and to obtain a standard form that is widely accepted. When working with fractions that include square roots in the denominator, multiplying both the numerator and the denominator by a suitable radical is essential.
Here's how it works:

• Find the radical present in the denominator.
• Multiply both the numerator and the denominator by the same radical.
• This process removes the square root from the denominator.

For example, if you have \( \frac{\sqrt{2}}{\sqrt{7}} \), you multiply by \( \frac{\sqrt{7}}{\sqrt{7}} \). This results in \( \frac{\sqrt{14}}{7} \) where the denominator is no longer a square root.

Properties of Square Roots

Square roots have specific properties that make them very useful in mathematical operations. Understanding these properties allows for easier simplification and manipulation of expressions involving square roots.
Here are some key properties:

• Product Property: The square root of a product is the same as the product of the square roots, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• Quotient Property: The square root of a quotient is the quotient of the square roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• Simplifying Roots: You can simplify square roots by dividing out perfect squares.

These properties assist in breaking down complex radicals and make calculations more straightforward. In our exercise, the quotient \( \sqrt{\frac{2}{7}} \) was separated into \( \frac{\sqrt{2}}{\sqrt{7}} \), demonstrating the quotient property.

Simplifying Fractions

Simplicity is key in fractions as it aids in clear understanding and easier computation. Simplifying a fraction means reducing it to its smallest form by dividing the numerator and the denominator by their greatest common factor (GCF).
While working with radicals, fractions may arise, and simplifying them follows the same basic principles.
Consider these steps:

• Identify the GCF of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• For radicals, ensure that no further simplification inside square roots is possible.

In the exercise, \( \frac{\sqrt{14}}{7} \) is already fully simplified. There is no common factor between 14 and 7 other than 1, meaning the expression is in its simplest radical form.