Problem 26 - College Algebra with Corequisite Support

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

We start by simplifying the fraction under the square root, \(\frac{405}{324}\). Find the greatest common divisor (GCD) of 405 and 324. The GCD is 81. Therefore, we can write \(\frac{405}{324} = \frac{405 \div 81}{324 \div 81} = \frac{5}{4}\).

Step 2: Apply Square Root to Numerator and Denominator

Next, we apply the square root to both the numerator and the denominator separately: \(\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{\sqrt{4}}\). Remember that \(\sqrt{4} = 2\).

Step 3: Final Simplification

Since \(\sqrt{5}\) cannot be simplified further, the expression becomes \(\frac{\sqrt{5}}{2}\). Therefore, the simplified form of the expression is \(\frac{\sqrt{5}}{2}\).

Key Concepts

Understanding the Greatest Common Divisor

To simplify any fraction, you need to find a number that divides both the numerator and the denominator exactly, without leaving a remainder. This number is known as the Greatest Common Divisor (GCD). It helps to reduce fractions to their simplest form. Consider a fraction like \(\frac{405}{324}\). The method to find the GCD involves several steps.

• First, list the factors of both the numerator and the denominator. For example, some factors of 405 include 1, 3, 5, 9, 15, 27, 45, 81, 135, and 405. Factors of 324 include 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, and 324.
• Next, identify the largest factor that is common between these two lists. In this case, the largest shared factor is 81. Therefore, 81 is the GCD of 405 and 324.

Finding the GCD is an essential skill in mathematics that aids in fraction simplification and problem-solving.

Demystifying Square Roots

Square roots can be an intriguing concept. The square root of a number is a value that, when multiplied by itself, gives the original number. Let’s apply this idea to the expression \(\sqrt{\frac{5}{4}}\). You can simplify the calculation by taking the square root of the numerator and the denominator separately.

• For the denominator, the square root of 4 is 2 because \(2 \times 2 = 4\).
• For the numerator, \(\sqrt{5}\) remains \(\sqrt{5}\) since 5 is not a perfect square, meaning it cannot be expressed as \(n \times n\) for any integer \(n\).

Knowing how to work with square roots allows for converting expressions into more manageable forms and is widely used both in pure and applied mathematics.

Simplifying Fractions

Fraction simplification is all about reducing fractions to their lowest terms, where both the numerator and the denominator are integers and have no common divisor other than 1. After identifying the GCD, divide both the numerator and the denominator by this value. In the expression \(\frac{405}{324}\), since the GCD is 81, dividing both parts by 81 leads to the simplified fraction of \(\frac{5}{4}\).

• The numerator, 405, divided by 81 is 5.
• The denominator, 324, divided by 81 is 4.

Once simplified, fractions often reveal much simpler relationships and values that are easier to work with, successfully breaking down complex mathematical problems.