Simplifying Complex Fractions: A Step-by-Step Guide

As an excellent high school mathematics teacher, I'm excited to share a detailed guide on simplifying the complex expression: \frac{\frac{4}{x} - \frac{4}{y}}{\frac{3}{x^2} - \frac{3}{y^2}}.

This type of expression can be challenging for many students, but with a systematic approach, you can master the art of simplifying complex fractions. Let's dive in!

Step 1: Understand the Structure

The given expression consists of two fractions: the numerator and the denominator. The numerator is \frac{4}{x} - \frac{4}{y}, and the denominator is \frac{3}{x^2} - \frac{3}{y^2}.

Step 2: Simplify the Numerator

To simplify the numerator, we need to find a common denominator for the two fractions and then subtract the fractions.

The common denominator is the least common multiple (LCM) of the denominators, which in this case is xy.

1. Convert the first fraction to the common denominator: \frac{4y}{xy}

2. Convert the second fraction to the common denominator: \frac{4x}{xy}

3. Subtract the fractions: \frac{4y}{xy} - \frac{4x}{xy} = \frac{4y - 4x}{xy}

Step 3: Simplify the Denominator

Similar to the numerator, we need to find a common denominator for the two fractions in the denominator and then subtract the fractions.

The common denominator is the LCM of the denominators, which in this case is x^2y^2.

1. Convert the first fraction to the common denominator: \frac{3y^2}{x^2y^2}

2. Convert the second fraction to the common denominator: \frac{3x^2}{x^2y^2}

3. Subtract the fractions: \frac{3y^2}{x^2y^2} - \frac{3x^2}{x^2y^2} = \frac{3y^2 - 3x^2}{x^2y^2}

Step 4: Combine the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final result:

\frac{\frac{4y - 4x}{xy}}{\frac{3y^2 - 3x^2}{x^2y^2}} = \frac{4y - 4x}{3y^2 - 3x^2}

This is the simplified form of the original expression.

Remember, the key to simplifying complex fractions is to focus on one step at a time, convert the fractions to a common denominator, and then perform the necessary operations. With practice, you'll become an expert at simplifying even the most challenging fraction expressions.

Happy learning!