Problem 52 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{3 a+6}{3}\)

Step-by-Step Solution

Answer

Question: Reduce the following rational expression to its simplest form: \(\frac{3a+6}{3}\). Answer: \(a+2\)

Step 1: Identify common factors of numerator and denominator

In the expression \(\frac{3a+6}{3}\), both the numerator and the denominator have a common factor of 3.

Step 2: Divide by the common factor

Now, divide the numerator and denominator by the common factor 3 to get the reduced expression. I.e., $$\frac{3a+6}{3} = \frac{3(a+2)}{3}$$

Step 3: Simplify the expression

As we have divided both numerator and denominator by the common factor, we can now simplify the expression: $$\frac{3(a+2)}{3} = a+2$$ So the reduced expression is \(a+2\).

Key Concepts

Simplifying Rational Expressions

A rational expression is similar to a fraction; it involves a numerator and a denominator. The goal of simplifying rational expressions is to reduce them to their simplest form, just like we simplify fractions.

To simplify a rational expression:

• First, determine the common factors in both the numerator and the denominator.
• Next, divide both parts by these common factors.
• Finally, write the simplified expression by removing the common factors.

In our exercise, the given rational expression was \( \frac{3a+6}{3} \).

This expression was simplified by identifying and removing the common factor of 3 from both the numerator and the denominator, resulting in the simplified expression \(a+2\). This means the expression cannot be simplified further.

Finding Common Factors

Finding common factors is a crucial step in simplifying rational expressions. It involves looking for numbers or variables that evenly divide both the numerator and the denominator.

Only factors that are common to both parts of the expression can be divided out to simplify the expression.

• In the expression \(3a+6\), the common factor is 3.
• We can factor 3 out of the numerator: \(3(a+2)\).
• The denominator here is also 3.

By dividing 3 out of both the top and bottom of the rational expression, we removed the shared factor, which simplifies the expression to \(a+2\). Remember, correctly identifying common factors is key to proper simplification.

Math Problem Solving

Solving math problems often requires breaking them down into smaller, more manageable steps. This approach makes complex problems simpler and less overwhelming. Follow these steps for effective math problem solving:

• Understand the problem: Read the problem carefully and identify what is being asked.
• Identify relationships: Look for patterns or commonalities in the expressions, such as shared factors in rational expressions.
• Simplify the problem: Divide the problem into smaller parts or rewrite expressions to make calculations easier.
• Check your work: After arriving at a solution, check your calculations for accuracy.

In our example, the solution involved identifying a common factor and simplifying the expression. By following these steps, we successfully reduced the expression to its simplest form of \(a+2\). Each math problem may present its unique challenges, but following a structured approach is important for solving them efficiently.