Problem 55 - Algebra and Trigonometry

Question

\(55-64=\) Simplify the compound fractional expression. $$ \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{x^{2}}-\frac{1}{y^{2}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-xy\).

Step 1: Simplify the Numerator

Consider the numerator \( \frac{x}{y} - \frac{y}{x} \). The first step is to find a common denominator, which in this case is \( xy \). Rewriting each term of the numerator with the common denominator, we have: \[ \frac{x^2}{xy} - \frac{y^2}{xy} = \frac{x^2 - y^2}{xy}. \] Thus, the simplified form of the numerator is \( \frac{x^2 - y^2}{xy} \).

Step 2: Simplify the Denominator

Consider the denominator \( \frac{1}{x^2} - \frac{1}{y^2} \). The common denominator here is \( x^2y^2 \). Rewriting each component with the common denominator results in: \[ \frac{y^2}{x^2y^2} - \frac{x^2}{x^2y^2} = \frac{y^2 - x^2}{x^2y^2}. \] Therefore, the denominator simplifies to \( \frac{y^2 - x^2}{x^2 y^2} \).

Step 3: Simplify the Fraction Expression

The overall expression becomes: \[ \frac{\frac{x^2 - y^2}{xy}}{\frac{y^2 - x^2}{x^2 y^2}}. \] Remembering the division of fractions rule \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \), the expression simplifies to: \[ \frac{x^2 - y^2}{xy} \times \frac{x^2 y^2}{y^2 - x^2}. \] Because \( x^2 - y^2 = -(y^2 - x^2) \), the expression simplifies to: \[ \frac{x^2 y^2}{xy} \times (-1). \]

Step 4: Final Simplification

Cancel \( xy \) from the numerator and the denominator: \[ \frac{x^2 y^2}{xy} \times (-1) = -xy. \] Therefore, the simplified form of the given compound fractional expression is \( -xy \).

Key Concepts

Simplification of Fractions

Simplifying fractions is about making the expression as simple as possible while still representing the same value. In mathematical terms, it means reducing the fraction until the numerator and denominator have no common factors left.

For compound fractions where both the numerator and denominator are fractions themselves, the simplification becomes a little more involved. You start by simplifying the expressions in both the top and bottom separately, before tackling the entire fraction.

Let's break it down with a clear approach:

• First, find the common denominator for both fractions involved and combine them.
• Simplify the algebraic terms to find if they can be reduced further.
• Finally, simplify the entire expression by cancelling out common terms, if possible.

Numerator and Denominator

In any fraction, the numerator and denominator play an integral role, dictating the fraction's value. The numerator is the number or expression written above the fraction line, indicating how many parts are considered. Meanwhile, the denominator, positioned below the line, shows the total parts into which something is divided.

Understanding them is crucial when simplifying complex fractions. For compound fractional expressions, the numerator and the denominator themselves might be fractions, making them more complicated.

• The numerator, such as in the expression \( \frac{x}{y} - \frac{y}{x} \), needs to be rewritten with a common denominator, enabling us to simplify the terms easier.
• The denominator \( \frac{1}{x^{2}} - \frac{1}{y^{2}} \) needs to follow a similar process to ensure the compound fraction can be managed effectively.

Common Denominator

A common denominator is essential when adding or subtracting fractions. It represents a mutual multiple for both denominators, allowing us to compare or combine the fractions directly.

For instance, in the problem where we have \( \frac{x}{y} - \frac{y}{x} \), the common denominator is \( xy \). This allows you to express both fractions with the same base:

• \( \frac{x^2}{xy} - \frac{y^2}{xy} \) becomes \( \frac{x^2 - y^2}{xy} \), simplifying the computation.

Similarly, for \( \frac{1}{x^2} - \frac{1}{y^2} \), the common denominator is \( x^2y^2 \), resulting in a unified expression:

• \( \frac{y^2}{x^2y^2} - \frac{x^2}{x^2y^2} \) simplifies to \( \frac{y^2 - x^2}{x^2y^2} \).

Understanding and applying common denominators makes simplifying these complex fractions methodical and clear.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving variables, numbers, and operation signs. These expressions can range from simple to complex structures, involving multiple operations and terms.

In simplifying compound fractional expressions like \( \frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{x^{2}} - \frac{1}{y^{2}}} \), understanding how to manipulate these algebraic expressions is key.

• First, recognize patterns or identities like difference of squares, which simplify terms. For example, \( x^2 - y^2 \) can be broken down into \((x - y)(x + y)\).
• Use these identities to factor expressions further, reducing them effectively when rewritten in terms common to both the numerator and denominator.

Handling these algebraic expressions correctly is essential for effectively simplifying not only the individual fractions within but also the entire compound expression.