Problem 10 - A Graphical Approach to College Algebra

Question

Write each rational expression in lowest terms. $$\frac{14 z^{3}}{6 z^{2}}$$

Step-by-Step Solution

Answer

\( \frac{7z}{3} \)

Step 1: Identify Common Factors

To simplify the rational expression \( \frac{14z^3}{6z^2} \), we first identify the common factors in the numerator and the denominator. The numerical coefficients 14 and 6 have a greatest common factor of 2. Both \( z^3 \) and \( z^2 \) also share \( z^2 \) as a common factor.

Step 2: Simplify Numerical Coefficients

Divide both the numerator and the denominator by their greatest common factor, 2. This gives: \( \frac{14}{2} = 7 \) and \( \frac{6}{2} = 3 \). This simplifies the expression to \( \frac{7z^3}{3z^2} \).

Step 3: Simplify Variable Terms

Simplify the expression by canceling the common variable factor of \( z^2 \) in both the numerator and the denominator. This results in \( \frac{z^{3-2}}{z^{2-2}} = \frac{z^1}{1} = z \). Therefore, the expression becomes \( \frac{7z}{3} \).

Step 4: Final Step: Write Simplified Expression

The simplified form of the rational expression \( \frac{14z^3}{6z^2} \) in lowest terms is \( \frac{7z}{3} \).

Key Concepts

Common Factors

Understanding common factors is crucial when simplifying rational expressions. In essence, a common factor is a number or variable that divides exactly into each term of an expression. For example, consider the expression \( \frac{14z^3}{6z^2} \). Both the numerator and the denominator have several factors.

• Numerical Factors: 14 can be factored into 2 and 7, and 6 can be factored into 2 and 3.
• Variable Factors: \( z^3 \) can be expressed as \( z \cdot z \cdot z \) and \( z^2 \) as \( z \cdot z \).

Identifying these factors helps us in the next step, which involves dividing both the numerator and the denominator by their common factors. Recognizing these shared components allows us to simplify our expression more efficiently and bring it closer to its simplest form.

Greatest Common Factor

The greatest common factor (GCF) of two numbers or expressions is the largest factor that exactly divides both. In simplifying rational expressions, finding the GCF is pivotal for reducing terms. Let's look at the expression \( \frac{14z^3}{6z^2} \).

• Numerical GCF: The GCF of 14 and 6 is 2, as 2 is the highest number that can divide both 14 and 6 without leaving a remainder.
• Variable GCF: For variables, we see that \( z^2 \) is the GCF for \( z^3 \) and \( z^2 \), since it is the highest power of \( z \) that is present in both.

By dividing both the numerator and the denominator by their GCFs, we can drastically simplify our expression. It results in smaller coefficients and simpler exponents, making the expression \( \frac{7z}{3} \). This emphasizes the importance of utilizing GCF in mathematical simplification.

Simplifying Numerical Coefficients

When simplifying rational expressions, dealing with numerical coefficients correctly optimizes the process. For the given expression \( \frac{14z^3}{6z^2} \), the coefficients are 14 and 6. To simplify these, we follow these steps:

• Identify the GCF, which is 2.
• Divide both coefficients by this GCF: \( \frac{14}{2} = 7 \) and \( \frac{6}{2} = 3 \).

This adjustment simplifies the overall expression by reducing it to \( \frac{7z^3}{3z^2} \). With smaller coefficients, further simplification steps are more manageable, ensuring a neat and concise final result, \( \frac{7z}{3} \). Understanding how to simplify numerical coefficients effectively is a foundational skill in algebra, allowing us to streamline complex expressions.