Problem 48 - Algebra 2

Question

Evaluate each expression. \(\sqrt{\frac{4}{9}}\)

Step-by-Step Solution

Answer

\( \frac{2}{3} \)

Step 1: Understand the Expression

We are given the expression \( \sqrt{\frac{4}{9}} \). This expression involves evaluating a square root of a fraction.

Step 2: Apply the Property of Square Roots and Fractions

Recall that the square root of a fraction \( \sqrt{\frac{a}{b}} \) can be written as \( \frac{\sqrt{a}}{\sqrt{b}} \). Thus, \( \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} \).

Step 3: Evaluate the Square Roots

Compute the square roots of the numerator and the denominator individually. We have \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \).

Step 4: Simplify the Fraction

Substitute the values from Step 3 into the fraction to obtain \( \frac{2}{3} \). This is the simplified result of the expression.

Key Concepts

Understanding Square Roots

Square roots are fundamentally about finding a number which, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because 2 multiplied by 2 is 4.
The notation used for square roots is the radical symbol (\(\sqrt{}\)). When handling expressions that involve square roots, it’s vital to identify whether you’re dealing with a perfect square. Perfect squares, when rooted, produce whole numbers. Examples include \(4, 9, 16,\) and so on.
In our exercise, we want the square root of 4 and 9, both perfect squares: \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\).

• Focus on identifying the root of small numbers to build a strong foundation.
• Recognizing perfect squares helps in quick calculations.

Once you master the square roots of basic numbers, evaluating more complex expressions becomes easier.

Handling Fractions

Fractions represent parts of a whole and are typically expressed as \(\frac{a}{b}\), with \(a\) as the numerator and \(b\) the denominator. Knowing how to manipulate fractions is key in evaluating expressions like \(\sqrt{\frac{4}{9}}\).
To work with a fraction under a square root sign, we need to split the operation between the numerator and the denominator. That means finding the square root of each part separately before simplifying further.
Here’s a quick process:

• Write the fraction under the radical as \(\sqrt{\frac{a}{b}}\).
• Convert into two separate roots: \(\frac{\sqrt{a}}{\sqrt{b}}\).
• Evaluate each square root individually.

Understanding this approach demystifies the evaluation process of fractional expressions involving radicals.

Simplifying Expressions

Simplifying expressions essentially means making them as concise and manageable as possible. It often requires reducing fractions, combining like terms, or factoring.
When faced with expressions like \(\sqrt{\frac{4}{9}}\), simplification becomes the final step after separating the square root of the numerator and the denominator.
Let’s iterate the steps taken to simplify this expression:

• Identify that the numerator 4 and the denominator 9 are both perfect squares.
• Calculate their square roots separately: \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\).
• Finally, simplify the expression to \(\frac{2}{3}\).

Mastering the art of simplification enhances understanding and assists in dealing with complex expressions efficiently, whether they involve square roots or not.