Problem 119 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{20}{81}} $$

Step-by-Step Solution

Answer

\( \frac{2 \sqrt{5}}{9} \)

Step 1 - Simplify the Square Root of the Numerator and Denominator Separately

The given expression is \( \sqrt{\frac{20}{81}} \). Rewrite it by separating the square roots of the numerator and the denominator: \( \sqrt{\frac{20}{81}} = \frac{\sqrt{20}}{\sqrt{81}} \).

Step 2 - Simplify the Denominator

Notice that \(81\) is a perfect square. Simplify \( \sqrt{81} \): \( \sqrt{81} = 9 \). Thus, the expression becomes \( \frac{\sqrt{20}}{9} \).

Step 3 - Simplify the Numerator

To simplify \( \sqrt{20} \), factor 20 into its prime factors: \( \sqrt{20} = \sqrt{4 \times 5} \). Knowing that \(4\) is a perfect square, simplify further: \( \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} \). Thus, \( \frac{\sqrt{20}}{9} \) becomes \( \frac{2 \sqrt{5}}{9} \).

Key Concepts

Square Root Simplification

Square root simplification helps break down complex roots into simpler, more manageable parts. It involves expressing the square root of a product as the product of the square roots. For example, consider the simplified square root in the step-by-step solution:

• Given: \( \sqrt{\frac{20}{81}} \)
• We break it into: \( \frac{\sqrt{20}}{\sqrt{81}} \)

This strategy is useful because it allows us to handle each part separately. Simplifying the denominator first is often easier since it has perfect squares. Afterwards, we can concentrate on the numerator.

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks, which are prime numbers. This step is crucial for simplifying square roots more effectively. Let’s revisit the process using the example provided:

• We start with the numerator: \( \sqrt{20} \)
• Prime factorize 20: \( 20 = 4 \times 5 \)
• Simplify further since 4 is a perfect square, \( \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} \)

By converting the number into its prime factors, we can more easily identify pairs of factors that simplify to whole numbers. This makes the process of simplifying the square root simpler and straightforward.

Rationalizing Denominators

Rationalizing the denominator involves converting a fraction into a form where the denominator is a rational number. While the current example does not require rationalization because the denominator is already simplified, it’s useful to know the concept. Here’s a basic method to handle cases where rationalization is needed:

• If we have \( \frac{1}{\sqrt{b}} \), multiply both numerator and denominator by \( \sqrt{b} \)
• This yields \( \frac{\sqrt{b}}{b} \), removing the square root from the denominator
• This process makes calculations easier and conforms to standard mathematical practices.

Understanding how to rationalize denominators is beneficial for handling more complex square root expressions in future problems.