Problem 12 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{\frac{25}{64}}\)

Step-by-Step Solution

Answer

\( \sqrt{\frac{25}{64}} = \frac{5}{8} \).

Step 1: Identify the Problem

The question asks to evaluate the square root of a fraction: \( \sqrt{\frac{25}{64}} \).

Step 2: Break Down the Fraction

Write \( \frac{25}{64} \) as \( \frac{a^2}{b^2} \) where \( a \) and \( b \) are whole numbers. Here, \( 25 = 5^2 \) and \( 64 = 8^2 \). So, \( \frac{25}{64} = \frac{5^2}{8^2} \).

Step 3: Apply Square Root to Numerator and Denominator

Use the property \( \sqrt{\frac{a^2}{b^2}} = \frac{\sqrt{a^2}}{\sqrt{b^2}} \). Applying this property, we get \( \sqrt{\frac{25}{64}} = \frac{\sqrt{5^2}}{\sqrt{8^2}} \).

Step 4: Calculate the Square Roots

Calculate \( \sqrt{5^2} = 5 \) and \( \sqrt{8^2} = 8 \). So, \( \sqrt{\frac{25}{64}} = \frac{5}{8} \).

Key Concepts

Fraction Simplification

When it comes to simplifying fractions, the goal is to reduce the fraction to its simplest form—making it as easy to work with as possible. Let's break this down:

Fractions consist of a numerator (the top number) and a denominator (the bottom number). Simplifying involves finding common factors or using properties to make the fraction less complex.

• Identify the largest common factor that divides both the numerator and the denominator.
• Divide both parts of the fraction by this common factor.

This process is known as simplifying or reducing the fraction. Although this wasn't directly used in this exercise, understanding it is fundamental when dealing with more complex fractions in the denominator or numerator.

In exercises like the one discussed, simplifying fractions is essential as it often prepares you for other operations, ensuring you work with the simplest possible expression early in the problem-solving process.

Properties of Square Roots

Square roots allow us to find a value that, when multiplied by itself, gives us the original number. Understanding the properties of square roots is key to working with them, especially inside fractions.

The main property relevant to this exercise is that square roots can be split over a fraction:

• \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\ \) - This property lets you take the square root of both the numerator and the denominator separately.

In our exercise, this property helps us find simpler expressions to deal with. The process can make calculations more straightforward and helps us see the solution clearly.

It's important to remember that the square root of a positive fraction will always be positive, provided both numerator and denominator are positive. Also, accurately applying this property requires breaking down the numerator and denominator into perfect squares, if possible, which simplifies the expression further.

Numerator and Denominator Manipulation

Manipulating the numerator and denominator correctly is crucial when solving fraction-based square root problems. The problem required separate handling of the numerator and denominator before applying the square root.

Steps to manipulate included:

• Express each number as a product of its prime factors where possible. For this exercise, noting that \(25 = 5^2\) and \(64 = 8^2\) made it easier to apply the square root property later.
• Using the property \(\sqrt{\frac{a^2}{b^2}} = \frac{\sqrt{a^2}}{\sqrt{b^2}}\) effectively simplified the expression by taking the square roots of the numerator and denominator separately.

By correctly forming the squares, we were able to demonstrate the outcome clearly—achieving a final result that reveals the simplified fraction after square root application.

Understanding this manipulation ensures accurate and efficient problem-solving, particularly in exercises involving square roots and fractions together.