Problem 357 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{3}{20}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{15}}{10} \)

Step 1: Express the Square Root of a Fraction

The given expression is \ \( \sqrt{\frac{3}{20}} \). Using the property of square roots: \ \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), rewrite the expression as \ \( \frac{\sqrt{3}}{\sqrt{20}} \).

Step 2: Simplify the Square Root in the Denominator

Simplify \( \sqrt{20} \) by expressing 20 as a product of its prime factors. \ \( 20 = 4 \times 5 \). Thus, \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \).

Step 3: Substitute Back into the Expression

Substitute \( \sqrt{20} \) with \( 2\sqrt{5} \) in the expression, resulting in \ \( \frac{\sqrt{3}}{2\sqrt{5}} \).

Step 4: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{5} \), leading to: \ \( \frac{\sqrt{3}}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3} \cdot \sqrt{5}}{2 \cdot 5} = \frac{\sqrt{15}}{10} \).

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots is crucial for larger algebraic manipulations. A square root, represented by the symbol \( \sqrt{} \), is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \). When simplifying square roots, we factor the number into its prime factors and look for pairs.

For example, simplifying \( \sqrt{20} \):
1. Breakdown into prime factors: \( 20 = 4 \times 5 \)
2. Recognize that \( \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \)
Doing this makes the expression easier to work with.

Properties of Square Roots

Square roots have several properties that are helpful:

• The square root of a product: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property lets us break down complex expressions.
• The square root of a fraction: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This is useful for rationalizing denominators.
• The square root of a perfect square is just that number: \( \sqrt{n^2} = n \).

These rules let us manipulate and simplify square root expressions, making further calculations clearer.

Rationalization

Rationalization means to eliminate square roots from the denominator of a fraction. This is essential because we often prefer denominators to be integers for easier interpretation. For example, to rationalize \( \frac{\sqrt{3}}{2\sqrt{5}} \), follow these steps:
1. Multiply both the numerator and the denominator by \( \sqrt{5} \).
2. The expression becomes \( \frac{\sqrt{3} \times \sqrt{5}}{2 \times 5} = \frac{\sqrt{15}}{10} \).
This results in a fraction where the denominator is a rational number, simplifying further usage in equations or expressions.