Problem 59 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{\frac{144}{324 d^{2}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{3d} \).

Step 1: Simplify the Expression Inside the Radical

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

Step 2: Apply the Square Root

Next, apply the square root to each component of the fraction separately: \[ \sqrt{\frac{4}{9 d^2}} = \frac{\sqrt{4}}{\sqrt{9 d^2}} \]Compute the square root of 4, which is 2, and the square root of \(9 d^2\), which is \(3d\) (as \( \sqrt{9} = 3 \) and \( \sqrt{d^2} = d \)). Thus, the expression becomes: \[ \frac{2}{3d} \]

Step 3: Summary of Simplified Expression

After simplifying, the entire expression \( \sqrt{\frac{144}{324 d^2}} \) simplifies to \( \frac{2}{3d} \).

Key Concepts

Square Roots

The square root is a fundamental concept in mathematics, especially when it comes to simplifying expressions. Essentially, finding the square root of a number is the same as asking what number, when multiplied by itself, gives you the original number. For example, the square root of 4 is 2 because 2 multiplied by itself (2 x 2) equals 4.
In algebra, when dealing with expressions under a square root, we often try to simplify them by breaking them down into smaller, manageable parts. Helpful steps include:

• Identify perfect squares, where possible.
• Simplify numerical components before considering any variables.

When you have a fraction under a square root, like in this exercise, you can apply the square root separately to the numerator and the denominator, making calculation more straightforward.
In our example, applying the square root function separately made it easier to handle, resulting in a cleaner and simpler expression.

Fractions

Fractions are a representation of parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). Simplifying fractions is crucial for making problems easier to solve and expressions easier to understand. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
In the given exercise, we dealt with the fraction \( \frac{144}{324d^2} \). By simplifying it using the GCD, we transformed it into \( \frac{4}{9d^2} \). This simplification process involved dividing 144 and 324 by 36, turning a seemingly complex problem into a much simpler one.

• Simplification leads to easier mental calculations and cleaner results.
• Often, simplified fractions will offer more insight into the problem's structure.

Understanding fractions and how to simplify them is a key skill in both algebra and daily math use.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential tool in mathematics, particularly useful when simplifying fractions. The GCD of two numbers is the largest integer that divides both numbers without leaving a remainder.
To find the GCD:

• Determine the prime factors of each number.
• Identify the largest factor that both numbers share.

In our exercise, the GCD of 144 and 324 was determined to be 36. Dividing both numbers by this GCD helped simplify the expression significantly. This method is invaluable in reducing fractions to their simplest form, ensuring clarity and ease in subsequent calculations.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression as simple as possible. This involves reducing complex fractions, expanding brackets, and canceling out variables where possible to make an expression more manageable.
This simplification not only makes the expression easier to interpret but also makes further calculations more efficient. In our example:

• Simplifying the fraction inside the square root made it easier to apply the square root separately to the numerator and the denominator.
• This ultimately led us to simplify the entire expression to \( \frac{2}{3d} \).

By simplifying algebraic expressions, we gain clearer insight into the relationships between variables and the overall structure of the expression.