Problem 59 - College Algebra

Question

Simplify each expression. $$\sqrt{\frac{144}{324 d^{2}}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{2}{3d}\).

Step 1: Simplify the Fraction

Start by simplifying the fraction inside the square root: \(\frac{144}{324d^2}\). Find the greatest common divisor of 144 and 324, which is 36. Divide both the numerator and the denominator by 36.\[ \frac{144}{324} = \frac{144 \div 36}{324 \div 36} = \frac{4}{9} \]Thus, the fraction simplifies to \(\frac{4}{9d^2}\).

Step 2: Simplify the Square Root

Next, take the square root of the simplified fraction. Recall that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).So, apply this to \(\sqrt{\frac{4}{9d^2}}\).\[ \sqrt{\frac{4}{9d^2}} = \frac{\sqrt{4}}{\sqrt{9d^2}} = \frac{2}{3d} \]This results in the fully simplified expression.

Key Concepts

Square Root Simplification

Square root simplification is a crucial skill in mathematics, especially when dealing with fractions under square root symbols. Learning this concept allows you to simplify expressions and calculations. To simplify a square root involving a fraction, remember the property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This means you can take the square root of the numerator and the denominator separately.

• Identify each part of the fraction under the square root.
• Apply the square root to both the numerator and denominator individually.

In our example, after simplifying the fraction to \( \frac{4}{9d^2} \), we take \( \sqrt{4} = 2 \) and \( \sqrt{9d^2} = 3d \). Thus, \( \sqrt{\frac{4}{9d^2}} = \frac{2}{3d} \). This helps in simplifying calculations and is essential for handling more complex algebraic operations efficiently.

Greatest Common Divisor

The greatest common divisor (GCD) plays a significant role in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Here are some key points on how to find and use the GCD in simplification:

• List the factors of both numbers involved. In the case of 144 and 324, identify factors for each.
• Determine the largest factor that appears in both lists.

For instance, the GCD of 144 and 324 is 36. By dividing both numbers by their GCD, we simplify \( \frac{144}{324} \) to \( \frac{4}{9} \).
Using the GCD is a powerful method to reduce fractions efficiently, making subsequent algebraic operations simpler and more manageable. Understanding how to utilize the GCD effectively can save time and reduce errors in calculations.

Fraction Simplification

Fraction simplification is essential in algebra for making complex expressions more manageable and easier to work with. Simplifying fractions involves reducing them to their simplest form. Here’s how you can simplify a fraction effectively:

• Find the greatest common divisor (GCD) of the numerator and the denominator, as explained previously.
• Divide both the numerator and the denominator by the GCD, ensuring they remain whole numbers if possible.

When dealing with algebraic fractions, such as \( \frac{144}{324d^2} \), simplifying the numerical part and the algebraic part separately makes it easier. After finding that the GCD is 36, the fraction becomes \( \frac{4}{9d^2} \).
This reduction helps in executing further arithmetic operations as it reduces the complexity of the expressions, leading to easier calculations and a better understanding of subsequent steps.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions follows the same basic principles as fraction simplification but often involves more complexity due to the presence of variables. To simplify a rational expression:

• Factorize both the numerator and the denominator completely, if necessary.
• Cancel any common factors to reduce the expression to its simplest form.

In our example, after fraction simplification, we treated \( \frac{4}{9d^2} \) as a rational expression. Simplifying involved recognizing \( 9d^2 \) as a perfect square, allowing further simplification under the square root.
Being proficient in dealing with rational expressions is crucial for solving equations and inequalities in algebra, as it simplifies problem-solving and enhances comprehension of algebraic structures.