Problem 55 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. $$ \sqrt{\frac{2}{7}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{14}}{7} \).

Step 1: Identify the fraction under the square root

We start by recognizing that the expression inside the square root, \( \frac{2}{7} \), is a fraction with a numerator of 2 and a denominator of 7.

Step 2: Apply the square root identity for fractions

We can simplify \( \sqrt{\frac{2}{7}} \) by using the identity \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Applying this identity, the expression becomes \( \frac{\sqrt{2}}{\sqrt{7}} \).

Step 3: Rationalize the denominator

To remove the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{7} \). This gives us \( \frac{\sqrt{2} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \), which simplifies to \( \frac{\sqrt{14}}{7} \) since \( \sqrt{7} \cdot \sqrt{7} = 7 \).

Key Concepts

Square Root Properties

When dealing with square roots, it's important to understand their fundamental properties. One of the key concepts is that the square root of a number represents a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

Square roots have a specific set of rules that make them easier to manipulate:

• Product Property of Square Roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
• Quotient Property of Square Roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

These properties allow you to break down more complex expressions into simpler parts. For instance, with the quotient property, the exercise simplifies \( \sqrt{\frac{2}{7}} \) to \( \frac{\sqrt{2}}{\sqrt{7}} \), making it easier to work with the expression.
Understanding these rules is crucial when rationalizing denominators, as they help you manipulate square roots in a systematic way.

Fractions in Radicals

Fractions can be present either under a square root, as in \( \sqrt{\frac{a}{b}} \), or as the result of a division involving square roots, such as \( \frac{\sqrt{a}}{b} \). When dealing with these types of expressions, the properties of square roots are often applied to simplify them.

The step of splitting the square root of a fraction into the quotient of square roots is particularly useful. For example:

• \( \sqrt{\frac{2}{7}} \) becomes \( \frac{\sqrt{2}}{\sqrt{7}} \)
• This step allows you to work with each number separately, which can simplify further calculations.

Furthermore, when fractions appear under a square root, you may encounter the need to rationalize them, especially if a square root ends up in the denominator. In such cases, multiplying the numerator and denominator by an appropriate radical can help release the square root, turning it into a more manageable integer or another rational number.

Multiplication of Radicals

Understanding how to multiply radicals is essential when simplifying expressions, particularly when rationalizing denominators. Radicals can be multiplied using the product property of square roots, which states that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \).

In the context of rationalizing denominators, this property allows you to effectively eliminate square roots from the denominator. Here's how it works:

• Given the expression \( \frac{\sqrt{2}}{\sqrt{7}} \), multiply both the numerator and denominator by \( \sqrt{7} \) to maintain the expression's equality.
• This results in \( \frac{\sqrt{2} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} \), or \( \frac{\sqrt{14}}{7} \).
• The product \( \sqrt{7} \times \sqrt{7} = 7 \) simplifies the denominator to an integer.

By converting the denominator into a rational number, the fraction becomes easier to handle and less complex to work with in further calculations. This is why understanding and using the multiplication of radicals is integral when handling such expressions.