Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{x+13}{x^{3}(3-x)} \cdot \frac{x(x-3)}{5}\)

Step-by-Step Solution

Answer

The final simplified form of the expression is \(\frac{x+13}{5x^{2}}\)

Step 1: Multiplication

Multiply the provided expressions. This multiplication follows a standard rule: Multiply the numerators (the top parts) and the denominators (the bottom parts) separately. Therefore, \[\frac{x+13}{x^{3}(3-x)} \cdot \frac{x(x-3)}{5} = \frac{(x+13) \cdot x(x-3)}{x^{3}(3-x) \cdot 5}\]

Step 2: Simplification

Now simplify the expression. Here, we have an \(x-3\) in both the numerator and denominator, and they would cancel out:\[\frac{(x+13) \cdot x(x-3)}{x^{3}(3-x) \cdot 5} = \frac{(x+13) \cdot x}{x^{3} \cdot 5}\]Further, there's a common factor of 'x' in both the numerator and denominator which also cancel out:\[\frac{(x+13) \cdot x}{x^{3} \cdot 5} = \frac{x+13}{x^{2} \cdot 5}\]This is the final simplified form of the expression, which can't be simplified anymore.

Key Concepts

Understanding Rational Expressions

Rational expressions are similar to fractions, but instead of just numbers, they involve variables. Think of them as fractions where the numerator and/or the denominator contain polynomials. For example, in the expression \(\frac{x+13}{x^{3}(3-x)}\), both the numerator and the denominator have polynomials.
To work with these expressions, remember a few basic rules:

• A rational expression is undefined if the denominator is zero, so always check for values that make the denominator zero.
• Factor expressions completely to identify common factors that might cancel out between the numerator and the denominator.

Understanding these basics helps you manipulate and simplify expressions like the one given above effectively.

Simplifying Expressions Made Easy

Simplifying an algebraic expression means reducing it to its simplest form. This involves:

• Factoring expressions fully.
• Cancelling common factors from the numerator and the denominator.

In our original problem, after multiplying, we had common factors of \((x-3)\) and \(x\) in both the numerator and denominator. These were cancelled out, leading us to a simplified expression of \(\frac{x+13}{x^{2} \cdot 5}\).
Remember, only identical factors can be cancelled! Always ensure everything is correctly factored out first, and watch out for common algebraic traps like cancelling terms that aren’t factors.

Mastering the Multiplication of Fractions

Multiplying fractions simply involves multiplying straight across the numerators and denominators. For algebraic fractions, this means multiplying the polynomials in the numerators and those in the denominators separately. In our example:

• Numerator: Multiply \((x+13)\) by \(x(x-3)\).
• Denominator: Multiply \(x^3(3-x)\) by \(5\).

Once these steps are followed, the expression becomes more straightforward to simplify, as shown in the steps above.
Handling algebraic fractions requires care, especially in looking for like terms to simplify after multiplication. Efficient handling of polynomials helps maintain clarity and accuracy in your answers.