Problem 26 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{405}{324}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt{5}}{2} \).

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{405}{324} \). Divide the numerator and the denominator by their greatest common divisor (GCD). The GCD of 405 and 324 is 81. Thus, \( \frac{405}{324} = \frac{405 \div 81}{324 \div 81} = \frac{5}{4} \).

Step 2: Apply the Square Root to the Fraction

The expression is \( \sqrt{\frac{5}{4}} \). By the properties of square roots, this can be rewritten as \( \frac{\sqrt{5}}{\sqrt{4}} \).

Step 3: Simplify the Square Root of the Denominator

The denominator has a perfect square, so take the square root of 4, which is 2. This yields \( \frac{\sqrt{5}}{2} \).

Step 4: Write the Final Simplified Expression

The simplified form of \( \sqrt{\frac{405}{324}} \) is \( \frac{\sqrt{5}}{2} \).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is an essential concept when it comes to simplifying expressions. It represents the largest number that can divide two or more numbers without leaving a remainder. Here’s why it's important:

• Finding the GCD helps reduce fractions to their simplest form, making them easier to work with.
• The GCD keeps numbers manageable by eliminating any unnecessary complexity from the fraction.

For example, consider the numbers 405 and 324. The GCD of these two numbers is 81. This means that 81 is the largest number that can evenly divide both 405 and 324.To find the GCD, you can use several methods such as listing divisors, prime factorization, or the Euclidean algorithm, which involves repeated division. Once you identify the GCD, you can divide both the numerator and the denominator of the fraction by it to achieve the simplest form, just as it was done by dividing both 405 and 324 by 81 to get \( \frac{5}{4} \). This simplification makes further mathematical operations more straightforward.

Square Roots

Understanding square roots is crucial in simplifying expressions, as they often appear in various mathematical problems. The square root of a number is a value that, when multiplied by itself, will give the original number.Here are some key aspects of square roots:

• A square root of \( n \) is typically represented as \( \sqrt{n} \).
• Square roots can simplify calculations and expressions by breaking down complex components.
• The square root of a perfect square will yield an integer, making calculations easier.

When you have a square root involving a fraction, like \( \sqrt{\frac{5}{4}} \), it can be separated into the square root of the numerator and the square root of the denominator. For instance, \( \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{\sqrt{4}} \).Here, \( \sqrt{4} \) simplifies to 2 because 4 is a perfect square. This simplification process is crucial for making expressions more manageable and for achieving a more straightforward final result.

Fractions

Fractions are a way of expressing parts of a whole and are widely used in mathematics to represent division. They consist of a numerator and a denominator, where the numerator is the part being divided by the denominator.Here are some important points about fractions:

• Simplifying fractions involves bringing them down to their lowest terms, making comparisons and computations easier.
• When simplifying, it is crucial to find the greatest common divisor (GCD) to reduce both the numerator and denominator.

In the exercise, the fraction \( \frac{405}{324} \) was simplified using its GCD, which gave us \( \frac{5}{4} \). This simplified fraction represents the same value but in a simpler form.Once you have a simpler fraction, any further mathematical operations, including taking square roots, are easier to manage. Fractions are often an intermediate but essential part of many larger mathematical procedures, so understanding how to manipulate them is a fundamental skill.