Problem 22 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{96}{100}}$$

Step-by-Step Solution

Answer

\( \frac{2\sqrt{6}}{5} \)

Step 1: Simplify the Fraction Inside the Square Root

Identify the fraction inside the square root: \( \frac{96}{100} \). This fraction can be simplified by finding the greatest common divisor (GCD) of 96 and 100, which is 4. Divide the numerator and the denominator by the GCD: \( \frac{96 \div 4}{100 \div 4} = \frac{24}{25} \).

Step 2: Apply the Square Root to the Fraction

The expression is now \( \sqrt{\frac{24}{25}} \). The square root of a fraction \( \frac{a}{b} \) can be expressed as \( \frac{\sqrt{a}}{\sqrt{b}} \). Therefore, \( \sqrt{\frac{24}{25}} = \frac{\sqrt{24}}{\sqrt{25}} \).

Step 3: Simplify the Square Roots

Calculate \( \sqrt{25} \), which equals 5 since 25 is a perfect square. For \( \sqrt{24} \), simplify by expressing 24 as a product of its factors: 24 = 4 * 6. Since 4 is a perfect square, \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \).

Step 4: Construct the Simplified Expression

Substituting back the simplified square roots, the expression \( \frac{\sqrt{24}}{\sqrt{25}} \) becomes \( \frac{2\sqrt{6}}{5} \). Thus, the simplified form of \( \sqrt{\frac{96}{100}} \) is \( \frac{2\sqrt{6}}{5} \).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial tool in simplifying fractions. It represents the largest integer that divides both the numerator and the denominator without leaving a remainder. For example, when simplifying the fraction \( \frac{96}{100} \), we find the GCD of 96 and 100. To determine this, list the factors of each number:

• 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
• 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

As we can see, the largest common factor is 4. This means 4 is the greatest common divisor of 96 and 100. By dividing both the numerator and the denominator by their GCD, we simplify the fraction to \( \frac{24}{25} \). Using the GCD ensures that the fraction is in its simplest form, making it easier to handle in further calculations.

Square Root

The square root of a number is another number which, when multiplied by itself, gives the original number. When dealing with fractions under a square root, it's essential to simplify the fraction first, as we did from \( \frac{96}{100} \) to \( \frac{24}{25} \). Then, we apply the square root to both the numerator and the denominator, separately. This is expressed as \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Eventually, we take each part: for \( \frac{24}{25} \), we need \( \sqrt{24} \) and \( \sqrt{25} \). Separating the operation makes the process simpler, helping ensure accuracy in calculations. Despite the complexity of some numbers like 24, with a bit of factorization, it becomes manageable.

Perfect Square

A perfect square is a number that has a whole number as its square root. Recognizing perfect squares is helpful when simplifying square roots because it allows us to directly replace them with their square root, simplifying calculations. In this exercise:

• 25 is a perfect square, since \( 5 \times 5 = 25 \), making \( \sqrt{25} = 5 \).
• Interestingly, 24 is not a perfect square, but it can be expressed in terms of perfect squares, for instance, \( 24 = 4 \times 6 \). This allows us to manipulate its square root as \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \).

Identifying and using perfect squares efficiently reduces computation steps and simplifies the expressions further.

Fraction Simplification

Fraction simplification is a process of reducing the fraction to its simplest form. It involves dividing both the numerator and the denominator by their greatest common divisor.

Once simplified, it becomes easier to apply operations like taking square roots, as we did with \( \sqrt{\frac{24}{25}} \). This streamlined process involves:

• Calculating the GCD, as seen where we simplified \( \frac{96}{100} \) to \( \frac{24}{25} \).
• Applying mathematical operations efficiently, such as computing \( \sqrt{24} \) with factorization, resulting in \( 2\sqrt{6} \).

Fraction simplification is beneficial not just for clarity but also for facilitating further mathematical operations, leading to clean, understandable results like \( \frac{2\sqrt{6}}{5} \). It saves time and enhances comprehension by reducing complexity in expressions.