Problem 24 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{3 a+3}{3}$$

Step-by-Step Solution

Answer

The reduced expression is \(a + 1\).

Step 1: Identify the common factors

First, look at the numerator \(3a + 3\) and the denominator \(3\). Identify any common factors that can be factored out from both the numerator and the denominator.

Step 2: Factor the numerator

The numerator \(3a + 3\) has a common factor of \(3\). Factor out the \(3\): \[ 3a + 3 = 3(a + 1) \]

Step 3: Simplify the fraction

Rewrite the rational expression with the factored numerator: \[ \frac{3(a + 1)}{3} \] Now, since \(3\) is common in both the numerator and the denominator, they can be canceled out: \[ \frac{3(a + 1)}{3} = a + 1 \]

Key Concepts

Factoring

Factoring is a key step in simplifying rational expressions. It involves breaking down a complex expression into simpler components that are multiplied together.

In the exercise provided, we start with the rational expression \( \frac{3a + 3}{3} \). The first step involves factoring the numerator.

Here, the numerator is \( 3a + 3 \). Both terms in the numerator have a common factor of 3. By factoring out 3, the expression becomes \( 3(a + 1) \).

Remember that identifying common factors and factoring expressions correctly is essential for simplifying rational expressions.

Canceling Common Factors

Once the expressions have been factored, the next step involves canceling out the common factors from the numerator and the denominator. This is a crucial step to simplify the rational expression.

In our example, after factoring out, we arrived at \( \frac{3(a + 1)}{3} \). Notice that 3 is a common factor in both the numerator and the denominator. Since they are the same, they can be canceled out.

To cancel, we simply remove the common factor from both places which simplifies the expression to \( a + 1 \).

This step reduces the complexity of the expression, making it much simpler and cleaner.

Numerator and Denominator

Understanding the numerator and denominator is fundamental to simplifying rational expressions. The numerator is the top part of the fraction, and the denominator is the bottom part.

In our example, the numerator is \( 3a + 3 \) and the denominator is \( 3 \). First, focus on simplifying the numerator by factoring. Secondly, look for any common factors between the numerator and denominator that can be canceled.

Always make sure to recheck the original expression after simplification to verify that it’s in its simplest form.

By clearly identifying and processing the numerator and denominator separately, you can effectively simplify any rational expression. This ensures you achieve the most streamlined version of the expression.