Problem 55 - 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 Problem

In the expression \( \sqrt{\frac{2}{7}} \), the denominator is under a square root, specifically \( \sqrt{7} \). Our task is to rationalize this denominator.

Step 2: Multiply Numerator and Denominator by \( \sqrt{7} \)

To rationalize \( \sqrt{\frac{2}{7}} \), multiply both the numerator and the denominator by \( \sqrt{7} \) to eliminate the square root in the denominator. This gives us: \[ \sqrt{\frac{2}{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{14}}{7} \].

Step 3: Simplify the Expression

The denominator \( \sqrt{7} \times \sqrt{7} \) becomes \( 7 \) because the square root of a number times itself equals the original number. The expression thus simplifies to: \( \frac{\sqrt{14}}{7} \). No further simplification is possible, so this is our final answer.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that represent a number which, when multiplied by itself, results in the original number under the root. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. When we deal with expressions like \( \sqrt{\frac{2}{7}} \), the square root applies to both the numerator and the denominator.

In operations involving square roots, it's important to remember that they must be non-negative for real numbers. Negative results under square roots are not part of the real number system. But when square roots are involved in a fraction, as in the exercise, we encounter an additional step: rationalizing the denominator, which leads us to our next discussion.

Numerator and Denominator Multiplication

Multiplying the numerator and denominator of a fraction is a common technique used in rationalizing denominators. When you have a square root in the denominator, like \( \sqrt{7} \) in our exercise, the goal is to eliminate it. This is done by multiplying both the top and bottom of the fraction by the same square root.

• The multiplication \( \sqrt{\frac{2}{7}} \times \frac{\sqrt{7}}{\sqrt{7}} \) ensures the fraction's value doesn’t change, because we are effectively multiplying by 1.
• This process results in the new expression \( \frac{\sqrt{14}}{7} \), where the denominator is now a rational number instead of a square root.

This multiplication method is essential as it allows us to simplify the fraction, preparing it for further simplification or making it easier to work with in other calculations.

Simplification of Radicals

The simplification of radicals involves rewriting a square root or radical expression in its simplest form. This process ensures that our mathematical expressions are as straightforward as possible. In our exercise, once the multiplication of the numerator and denominator is complete, we have \( \frac{\sqrt{14}}{7} \).

Here, the denominator is no longer a radical, having been rationalized to 7. The numerator consists of \( \sqrt{14} \), which is already in its simplest form because 14 has no perfect square factors other than 1.

• Simplifying involves checking the numerator, \( \sqrt{14} \), ensuring it cannot be broken down further into any simpler square root form.
• The operation also makes the fraction easier to understand and work with in both theoretical math and practical applications.

The process we've completed is typical when addressing radicals in the denominator and ensures clarity and simplicity in our math expressions.