Problem 37 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{24 n^{4}}{16 n^{4}+24 n^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3n}{2n + 3}\).

Step 1: Factor Out the Greatest Common Factor (GCF) in the Denominator

First, we need to examine the denominator \(16n^4 + 24n^3\) and factor out the greatest common factor. This GCF is \(8n^3\), so we get:\[16n^4 + 24n^3 = 8n^3 (2n + 3)\]

Step 2: Simplify the Rational Expression

Now that we've expressed the denominator in its factored form, rewrite the rational expression. It becomes:\[\frac{24n^4}{8n^3(2n + 3)}\]Next, factor the numerator, choosing the common factor between the numerator and the denominator. The GCF for the numerator is \(8n^3\). Factoring \(24n^4\) gives:\[24n^4 = 8n^3 \cdot 3n\]So the expression simplifies to:\[\frac{8n^3(3n)}{8n^3(2n + 3)}\]

Step 3: Cancel the Common Factors

Since \(8n^3\) appears in both the numerator and the denominator, we can cancel this common factor:\[\frac{8n^3(3n)}{8n^3(2n + 3)} = \frac{3n}{2n + 3}\]

Key Concepts

Factoring

Factoring plays a vital role in simplifying rational expressions. It involves breaking down an expression into its simplest building blocks, known as factors. For the rational expression given, we need to focus on both the numerator and the denominator.

A good starting point in factoring is to look for any common factors that can be grouped together. For instance, in the original exercise, we have the expression \(16n^4 + 24n^3\). To factor it successfully, we should identify numbers or variables that can divide both terms evenly. By doing this, we make the expression less complex and easier to handle in later steps.

In summary, factoring breaks down expressions into components, revealing hidden structures and making simplification possible.

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is essential when factoring expressions. The GCF is the largest factor shared by two or more terms. In the context of our exercise, we identify the GCF in the denominator \(16n^4 + 24n^3\) as \(8n^3\).

How do we find the GCF? Start by identifying the greatest number that each term can be divided by. Then, look at the variables; in this case, we choose the lowest power of \(n\) that appears in each term, which is \(n^3\). Combining these gives us the GCF \(8n^3\).

After finding the GCF, factor it out from the expression. This creates simpler sub-expressions to work with and makes the simplification process smoother.

Simplification

Simplification is the process of reducing expressions to their simplest form. After factoring, the goal is to cancel out any common factors between the numerator and the denominator.

Take the expression \(\frac{24n^4}{16n^4 + 24n^3}\). By factoring, we get \(\frac{8n^3 \cdot 3n}{8n^3 (2n + 3)}\). Notice the \(8n^3\) in both the numerator and the denominator. Since they are the same, they "cancel out", leaving us with \(\frac{3n}{2n + 3}\).

Simplification allows us to see the essence of the rational expression without any unnecessary complexity. It's a powerful tool that helps turn a complicated problem into an understandable one.