Problem 26 - College Algebra

Question

For the following exercises, 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, identify the greatest common divisor (GCD) of the numbers in the fraction 405 and 324. The GCD here is 81. Divide both the numerator and the denominator by 81:\[\frac{405}{324} = \frac{405 \div 81}{324 \div 81} = \frac{5}{4}\]

Step 2: Separate the Square Roots

Now that the fraction is simplified, we can take the square root of the numerator and the denominator separately:\[\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{\sqrt{4}}\]

Step 3: Simplify the Denominator

The denominator, \(\sqrt{4}\), can be simplified as follows because 4 is a perfect square:\[\sqrt{4} = 2\]Therefore, the expression becomes:\[\frac{\sqrt{5}}{2}\]

Step 4: Final Simplified Expression

After simplifying the denominator, the expression \(\sqrt{\frac{405}{324}}\) simplifies to:\[\frac{\sqrt{5}}{2}\]

Key Concepts

Greatest Common Divisor

When simplifying fractions, finding the greatest common divisor, or GCD, is an essential step. The GCD is the largest number that can evenly divide two or more integers without leaving a remainder. This concept is crucial for reducing fractions to their simplest form.

• To find the GCD of two numbers, like 405 and 324, you can list out the factors of both numbers and choose the largest one they have in common.
• Another way to find the GCD is by using the Euclidean algorithm, which involves repeated division. You divide the larger number by the smaller one and use the remainder in the next division, continuing this process until the remainder is zero.
• The last non-zero remainder is the GCD.

In our case, the GCD of 405 and 324 is 81. This helps us simplify the fraction to \( \frac{5}{4} \). Finding the GCD is like cleverly "shrinking" the numbers; making them smaller but still equal to the original fraction.

Square Roots

Square roots are fundamental when simplifying expressions, especially those involving fractions. The square root of a number is a value that, when multiplied by itself, yields the original number. In mathematical terms, for a number \( x \), the square root is written as \( \sqrt{x} \).

• Square roots can be separated over a fraction: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• This separation helps simplify calculations, as shown in the solved exercise where \( \sqrt{\frac{5}{4}} \) is simplified to \( \frac{\sqrt{5}}{\sqrt{4}} \).
• Remember, the square root of a number is only applicable to non-negative values in real numbers.

Understanding how to separate square roots over fractions is a useful skill when simplifying complex expressions. It breaks down the process into more manageable parts.

Perfect Squares

Perfect squares are numbers that have whole numbers as their square roots. For instance, 4, 9, 16, and 25 are all perfect squares because their square roots are 2, 3, 4, and 5, respectively.

• Recognizing perfect squares allows you to simplify square roots easily. For example, since 4 is a perfect square, \( \sqrt{4} = 2 \).
• This step is pivotal in simplifying expressions because it eliminates the radical in the denominator. In the given exercise, this allowed us to change \( \frac{\sqrt{5}}{\sqrt{4}} \) to \( \frac{\sqrt{5}}{2} \).
• Knowing perfect squares by heart can save time and make solving such expressions faster and more intuitive.

Utilizing the concept of perfect squares is a strategic move when simplifying expressions involving square roots, ensuring the final result is in the simplest form.