Problem 28 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{x^{3}}{x^{3}-x^{2} y}$$

Step-by-Step Solution

Answer

\( \frac{x}{x - y} \)

Step 1: Factor the Denominator

To simplify the fraction \( \frac{x^3}{x^3 - x^2y} \), the first step is to factor the denominator. Notice that \( x^2 \) is common in the terms \( x^3 \) and \( x^2y \), so we can factor \( x^2 \) out: \[ x^3 - x^2y = x^2(x - y). \]

Step 2: Simplify the Fraction by Canceling Common Factors

With the factored denominator, the expression becomes: \[ \frac{x^3}{x^2(x - y)}. \] Recognize that there is a common factor of \( x^2 \) in both the numerator and the denominator. Cancel \( x^2 \) from both:\[ \frac{x^3}{x^2(x - y)} = \frac{x^2 \cdot x}{x^2(x - y)} = \frac{x}{x - y}. \]

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making them easier to work with and understand. In algebraic fractions like \( \frac{x^3}{x^3 - x^2 y} \), the goal is to reduce the expression by finding equivalent fractions that have smaller or more straightforward numbers.

Here's how to approach it:

• Identify whether there are any common factors in the numerator and denominator.
• Cancel these common factors to simplify the expression.
• Always check if further simplification is possible.

By simplifying, calculations become much easier to handle, making problem-solving quicker and more efficient.

Factoring

Factoring is an essential mathematical skill that involves expressing an algebraic expression as a product of its factors. In the given problem, we need to factor the denominator \( x^3 - x^2y \).

To factor:

• Look for common terms. Here, both terms have \( x^2 \).
• Factor out the greatest common factor, resulting in: \( x^3 - x^2y = x^2(x - y) \).
• This process turns a difficult expression into a simpler one.

Factoring helps in recognizing which terms can be simplified out, providing a clearer path to solving the equation.

Canceling Common Factors

Once the factors are identified, the next logical step is canceling common factors—a crucial part of simplifying fractions. From the factored equation \( \frac{x^3}{x^2(x - y)} \), we notice \( x^2 \) appears in both the numerator and denominator.

To cancel common factors:

• Compare the terms both above and below the fraction line.
• Cross out these terms, ensuring the operation affects both the numerator and denominator equally.
• This leads us to the simpler fraction: \( \frac{x}{x - y} \).

This simplification step is vital as it reduces complexity and results in an expression that is easy to interpret and use.

Numerator and Denominator

Understanding the roles of the numerator and denominator is critical in handling fractions effectively. In any fraction, the numerator is the top part, while the denominator is the bottom part.
The numerator (\( x^3 \) in our problem) indicates how many parts we have, while the denominator (\( x^3-x^2y \) before factoring) tells us how many parts make up a whole.
When simplifying:

• Consider what each part represents.
• Think about how factoring affects the balance between the numerator and denominator.
• Remember that simplification does not change the actual value of the fraction, just its appearance.

Grasping these concepts helps demystify fractions, aiding both simple arithmetic and complex algebraic operations.