Problem 17 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{3+\frac{1}{x}}{3 x+1}}{x^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given complex rational expression: \(\frac{\frac{3x+\frac{1}{x}}{3x+1}}{x^2}\). Answer: \(\frac{1}{x^2}\)

Step 1: Rewrite the complex fraction

Eliminate the fraction within the fraction by multiplying both the numerator and denominator by x, to get rid of the \(\frac{1}{x}\): $$ \frac{\frac{3x+\frac{1x}{x}}{3x+1}}{x^2} $$ Now, the numerator of the complex fraction becomes: $$ \frac{3x+1}{3x+1} $$

Step 2: Simplify the complex fraction

Since the numerator and the denominator of the complex fraction is the same, we can simplify the complex fraction to 1: $$ \frac{1}{x^2} $$

Step 3: Write the final answer

The simplified complex rational expression is: $$ \frac{1}{x^2} $$

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This means that the numerator and the denominator have no common factors other than 1. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by their GCD.

In algebra, simplifying fractions often involves factoring expressions in both the numerator and the denominator. By identifying common factors, fractions can be reduced to their simplest form. This is useful because simplified expressions are easier to work with, especially when solving equations.

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. Just like numerical fractions, these algebraic terms can be simplified. Simplifying rational expressions involves:

• Factoring polynomials in the numerator and the denominator.
• Cancelling common factors from the numerator and the denominator.

It's important to understand that any value of the variable that makes the denominator zero is excluded from the set of valid solutions. Thus, while simplifying, keeping track of these values ensures the expression remains valid. Simplification helps to solve equations more efficiently, as seen in the provided exercise.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but contain variables in either the numerator, the denominator, or both. Simplifying these involves similar methods used for numeric fractions, along with the knowledge of polynomials. Here’s how to handle algebraic fractions:

• Factorize both the numerator and the denominator, similar to rational expressions.
• Use algebraic identities and common polynomial factors to simplify the fraction.

When simplifying, it's important to remove complex fractions, as shown in the exercise. A complex fraction has a fraction in either the numerator, the denominator, or both. By multiplying by appropriate terms, complex fractions can be converted into simpler, single-level fractions, making them easier to deal with. Understanding and mastering this concept is crucial for tackling more advanced algebraic problems.