Problem 15 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{9 c d^{2}}{12 c^{4} d^{2}}\)

Step-by-Step Solution

Answer

The simplest form is \( \frac{3}{4c^3} \), and it is undefined for \( c = 0 \).

Step 1: Identify Like Terms and Simplify

Begin simplifying the given rational expression by identifying and canceling like terms in the numerator and the denominator. The expression is \( \frac{9c d^{2}}{12c^{4} d^{2}} \). Notice that \( d^2 \) appears in both the numerator and denominator, allowing us to cancel them out directly, simplifying to \( \frac{9c}{12c^4} \).

Step 2: Simplify Further by Canceling Common Factors

To simplify \( \frac{9c}{12c^4} \), factor out the greatest common factor from the coefficients, which is 3. Thus, \( \frac{9}{12} \) simplifies to \( \frac{3}{4} \). Now address the \( c \) terms: \( c \) in the numerator simplifies with one of the \( c^4 \) in the denominator, resulting in \( c^{3} \) in the denominator. The expression becomes \( \frac{3}{4c^3} \).

Step 3: Determine Values Where the Expression is Undefined

For the simplified expression \( \frac{3}{4c^3} \), the fraction is undefined if the denominator equals zero. Set the denominator \( 4c^3 \) equal to zero: \( 4c^3 = 0 \). Solving for \( c \), you find \( c^3 = 0 \), so \( c = 0 \). Thus, the expression is undefined when \( c = 0 \).

Key Concepts

Understanding Undefined Expressions

In rational expressions, one critical aspect is identifying when an expression becomes undefined. This occurs when the denominator of the fraction equals zero because division by zero is not possible in mathematics. In the given example, we simplify the expression to \( \frac{3}{4c^3} \).

To determine when the expression is undefined, solve the equation \( 4c^3 = 0 \). Convert the problem into identifying the value of the variable that causes the denominator to be zero. Solving for \( c \), we find that \( c = 0 \) makes the denominator zero. Therefore, the expression is undefined for \( c = 0 \).

When working with rational expressions, always check for these undefined cases. It is essential for proper mathematical practice and ensures that computations are valid.

Canceling Common Factors in Rational Expressions

Simplifying rational expressions often involves canceling common factors present in both the numerator and the denominator. Consider the expression \( \frac{9c d^{2}}{12c^{4} d^{2}} \).

Firstly, identify the common factors across the numerator and denominator. In this example, \( d^2 \) is present in both, so we can cancel it out, simplifying the expression to \( \frac{9c}{12c^4} \).

Next, look at the coefficients: 9 and 12. Their greatest common factor is 3, so divide both by 3, resulting in \( \frac{3}{4} \).

For variable expressions such as \( c \), cancel one \( c \) from the numerator with one from the \( c^4 \) in the denominator, leaving \( c^3 \) in the denominator. Thus, achieving the simplest form \( \frac{3}{4c^3} \).

Canceling common factors is a vital step in simplifying rational expressions and makes further calculations easier.

Exploring Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying them involves reducing the fraction to its simplest form by canceling out common factors.

The example \( \frac{9c d^{2}}{12c^{4} d^{2}} \) illustrates the basic process of simplification:

• Identify like terms and factor out common elements.
• Simplify coefficients by their greatest common factor.
• Simplify polynomials by canceling common variables.

Rational expressions may seem daunting at first, but with practice, you can simplify them quickly. It's like solving little puzzles by breaking them down into smaller, more manageable pieces. By following these steps, simplifying and working with rational expressions becomes much more approachable. Understanding these expressions is crucial for algebraic problem-solving and helps in advancing to more complex mathematical topics.