Problem 7 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{\frac{9}{25}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{5} \).

Step 1: Rewrite the Square Root

Start by applying the property of square roots to rewrite the expression. The expression is \( \sqrt{\frac{9}{25}} \). Using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), rewrite the expression: \[ \sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}} \]

Step 2: Evaluate the Square Roots

Evaluate the square roots of the numerator and the denominator separately. - The square root of 9: \( \sqrt{9} = 3 \), because \(3 \times 3 = 9\).- The square root of 25: \( \sqrt{25} = 5 \), because \(5 \times 5 = 25\).

Step 3: Simplify the Fraction

Now replace the square roots in the fraction with their evaluated values. Thus, the expression becomes:\[ \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5} \]

Key Concepts

Square Roots

Understanding square roots is crucial in simplifying expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. Similarly, square roots play a role in both integers and fractions.

To simplify expressions involving square roots, such as \( \sqrt{\frac{9}{25}} \), we use the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This means the square root of a fraction can be separated into the square root of the numerator and the square root of the denominator.

• For the numerator 9: \( \sqrt{9} = 3 \)
• For the denominator 25: \( \sqrt{25} = 5 \)

This approach simplifies the fraction square root into manageable parts.

Fraction Simplification

Simplifying fractions involves expressing a fraction in its simplest form, which means the numerator and denominator are the smallest possible whole numbers that maintain the same value. When simplifying the fraction \( \frac{3}{5} \), derived from \( \frac{\sqrt{9}}{\sqrt{25}} \), it's already in its simplest form because 3 and 5 have no common factors other than 1.

Fraction simplification also involves understanding equivalent fractions. Equivalent fractions are different representations of the same part of a whole. A fraction is simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). However:

• In our expression, the fraction \( \frac{3}{5} \) is already simplified.
• 3 and 5 are prime to each other, meaning they don't share any divisor except 1.

Thus, no further simplification is possible.

Algebraic Properties

Algebraic properties help us manage expressions and equations involving variables and numbers more effectively. Important properties utilized in our problem include the properties of square roots and fractions.

One key property is the "product of square roots," which asserts that the square root of a product is equal to the product of the square roots of each number individually. Thus, \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \). This property was used when breaking down the square root of a fraction into separate square roots for the numerator and denominator in the expression \( \sqrt{\frac{9}{25}} \).

• Square root property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
• Simplification: evaluating each square root separately before simplifying fractions.

Using these properties, expressions can be broken down into simpler components, making them easier to evaluate and understand.