Problem 38 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{18 m^{4}}{36 m^{4}-9 m^{3}} $$

Step-by-Step Solution

Answer

\(\frac{2m}{4m - 1}\)

Step 1: Identify the Greatest Common Factor in the Numerator

Examine the numerator, which is \(18m^4\). The greatest common factor is \(18m^4\) because it is the only term present.

Step 2: Identify the Greatest Common Factor in the Denominator

Look at the denominator, \(36m^4 - 9m^3\). Find the greatest common factor. Both terms contain \(9m^3\), so the greatest common factor is \(9m^3\).

Step 3: Factor the Denominator

Factor out the greatest common factor from the denominator: \[ 36m^4 - 9m^3 = 9m^3(4m - 1) \]

Step 4: Write the Rational Expression with Factored Denominator

The expression now looks like this: \[ \frac{18m^4}{9m^3(4m - 1)} \]

Step 5: Simplify the Fraction

Divide both the numerator and the denominator by the common factor \(9m^3\). This gives: \[ \frac{18m^4 \div 9m^3}{9m^3(4m - 1) \div 9m^3} = \frac{2m}{4m - 1} \]

Step 6: Final Simplified Expression

The rational expression \(\frac{18m^4}{36m^4-9m^3}\) simplifies to \(\frac{2m}{4m - 1}\).

Key Concepts

Greatest Common Factor

When simplifying rational expressions, the first step often involves identifying the greatest common factor (GCF). This means finding the largest expression that divides all terms within a polynomial, both in the numerator and the denominator. Determining the GCF helps simplify the expression by reducing it to its simplest form.

For example, consider the rational expression:

• Numerator: \(18m^4\)
• Denominator: \(36m^4 - 9m^3\)

In the numerator, the GCF is straightforwardly the term itself, \(18m^4\), since it consists of a single term. In contrast, the denominator consists of two terms: \(36m^4\) and \(-9m^3\). For these terms, the GCF is \(9m^3\), because it is the largest term that can be factored out from both coefficients and the variables.

Finding the GCF is crucial because it makes subsequent steps like factoring more streamlined. It decreases complexity while maintaining the mathematical integrity of the expression.

Factoring Polynomials

Once you have determined the GCF, the next critical step is factoring the polynomial where applicable. This mainly applies to the denominator in most rational expressions. By factoring, you are essentially reverse-engineering the expression to see the components that make up the polynomial.

Let's take a closer look at the denominator from our example:

• The denominator \(36m^4 - 9m^3\) can be factored by taking out the \(9m^3\).
• This results in \(9m^3(4m - 1)\).

Factoring helps expose the inner structure of the polynomial, making it easier to handle for further simplification. In this case, removing the GCF separates the polynomial into two more straightforward expressions within parentheses.

The purpose of factoring is not only to make simplification possible but also to reveal cancellations between the numerator and denominator later in the process. Without this crucial step, simplification would be extremely difficult, if not impossible.

Simplification Steps

After factoring, the next big task is simplifying the rational expression by canceling common factors in the numerator and the denominator.

Let's see how this works in our example:

• The expression is rewritten with a factored denominator: \(\frac{18m^4}{9m^3(4m - 1)}\).
• You identify the common factor \(9m^3\), which appears in both the numerator and denominator.
• Divide both the numerator and the denominator by this common factor to simplify the expression.

This simplification is straightforward:\[\frac{18m^4 \div 9m^3}{9m^3(4m - 1) \div 9m^3} = \frac{2m}{4m - 1}\]

It's important to remember:

• Only like terms and common factors can be canceled.
• The expression is simplified, not solved.

The final simplified expression wouldn't be possible without assigning the GCF, factoring, and careful division of common factors. This process keeps the mathematical relationships intact and makes the expression easier to understand and work with.