Problem 65 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{x+3}{x-4} \cdot \frac{x-4}{x+1} \cdot \frac{x-2}{x+3} $$

Step-by-Step Solution

Answer

Question: Multiply the fractions and simplify the result: \(\frac{(x+3)}{(x-4)}\cdot\frac{(x-4)}{(x+1)}\cdot\frac{(x-2)}{(x+3)}\) Answer: \(\frac{x-2}{x+1}\)

Step 1: Multiply the numerators

To do this, simply multiply the terms inside the numerators of each fraction together: \((x+3)(x-4)(x-2)\).

Step 2: Multiply the denominators

Similarly, multiply the terms inside the denominators of each fraction together: \((x-4)(x+1)(x+3)\).

Step 3: Write the resulting fraction

Now, write the resulting fraction with the numerator found in Step 1 and the denominator found in Step 2: $$ \frac{(x+3)(x-4)(x-2)}{(x-4)(x+1)(x+3)} $$

Step 4: Simplify the fraction

Look for any common factors in the numerator and denominator and cancel them out. In this case, we have \((x+3)\) and \((x-4)\) as common factors: $$ \frac{\cancel{(x+3)}(x-4)(x-2)}{\cancel{(x-4)}(x+1)(x+3)} = \frac{(x-2)}{(x+1)} $$

Step 5: State the final simplified fraction

The fraction after simplification is: $$ \frac{x-2}{x+1} $$

Key Concepts

Fraction Simplification

When dealing with rational expressions, **fraction simplification** is a crucial step that makes expressions easier to understand and work with. Simplification involves reducing the expression to its simplest form. This usually means canceling out any common factors that appear both in the numerator and the denominator.

Here are a few tips to keep in mind when simplifying fractions:

• Always perform operations such as multiplications before starting the simplification.
• Look for terms that are exactly the same in both the numerator and the denominator.
• Cancel any matching terms since their ratio is one, leaving the fraction simplified.

After these steps, you'll have a much simpler expression, making future calculations less prone to errors.

Numerators and Denominators

Every rational expression is composed of a **numerator** (the term above the fraction line) and a **denominator** (the term below the fraction line). Understanding these components is fundamental to mastering fraction operations.

**Numerators** represent the "top" part of the fraction and are the terms being divided by the denominators. In a multiplication of fractions, multiply all numerators to form a new numerator.

**Denominators** form the "bottom" of the fraction, showing how many parts are being considered. When multiplying fractions, the denominators must be multiplied together to make a new denominator.

When simplifying, they play a crucial role. Identifying common terms within them allows for easy reduction of the fraction, ensuring you only have essential parts of the expression left.

Common Factors Cancellation

A key step in simplifying rational expressions is the **common factors cancellation** process. This involves removing identical factors from both the numerator and the denominator because their ratio is \(1\), which does not affect the overall value of the expression.

Here's how it works:

• After multiplying all numerators and denominators, examine both for shared terms.
• If you find a matching term, you can "cancel" it by crossing it out from both the numerator and the denominator.
• This results in a more streamlined expression, leaving only the necessary components.

This simplification step not only cleans up the expression but also makes it much easier to handle in subsequent calculations or real-world applications.