Problem 27 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{12 a^{3}}{18 a} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2a^2}{3} \).

Step 1: Identify Common Factors

First, look at the numerator \(12a^3\) and the denominator \(18a\) of the fraction. Identify the greatest common factor (GCF) of the numerical coefficients (12 and 18). The GCF is 6. Also, note the variable \(a\) which appears in both the numerator and the denominator.

Step 2: Factor the Expression

Factor out the GCF from both the numerator and the denominator. This gives us:\[\frac{12a^3}{18a} = \frac{6 imes 2a^3}{6 imes 3a}\]

Step 3: Cancel the Common Factors

Cancel out the common factors from the numerator and the denominator. Here, cancel the 6s and one \(a\), as these are common to both the numerator and the denominator:\[\frac{6 imes 2a^3}{6 imes 3a} = \frac{2a^{3-1}}{3}\]Upon simplifying the exponent of \(a\), we get \(a^{3-1} = a^2\).

Step 4: Write the Simplified Expression

After canceling the common factors and simplifying, the expression reduces to:\[\frac{2a^2}{3}\]

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is crucial when simplifying rational expressions. The GCF is the largest factor that divides two or more numbers or terms. In our exercise, we have the rational expression \( \frac{12a^3}{18a} \). To find the GCF of the numerical coefficients 12 and 18, list their factors:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

From the lists, the largest common factor is 6. Similarly, for the variables, identify the lowest power of \(a\) that appears in both the numerator and the denominator, which in this case is \(a\). Finding the GCF helps us to simplify expressions by reducing terms to their simplest form. This is the first step in tackling tasks like the one in our exercise.

Factoring Expressions

Factoring is the process of breaking down an expression into products of simpler expressions. After identifying the GCF in our rational expression \( \frac{12a^3}{18a} \), it’s important to factor it out from both the numerator and the denominator. We do this by expressing both 12 and 18 in terms of their GCF as multiples:

• 12 can be written as \(6 \times 2\)
• 18 can be written as \(6 \times 3\)

Additionally, you factor out \(a\) from the variable terms, which gives us: \[ \frac{6 \times 2a^3}{6 \times 3a} \] This factoring step prepares the expression for simplification by visibly highlighting the common factors in both the numerator and denominator that can be canceled.

Canceling Common Factors

After factoring the numerator and the denominator, the next step is canceling out the common factors. This is the most satisfying part because it simplifies the expression effectively. In the factorized version of our expression \( \frac{6 \times 2a^3}{6 \times 3a} \), both the numerator and the denominator contain the common factor 6 and the variable \(a\). Cancel these out:

• The 6s in the numerator and denominator cancel each other.
• The \(a\) in the numerator and denominator also cancel out, reducing the \(a^3\) in the numerator to \(a^{3-1} = a^2\).

The remaining terms give us the simplified result: \( \frac{2a^2}{3} \). It’s crucial to only cancel factors that appear in both the numerator and denominator to avoid altering the value or meaning of the expression.