Problem 43 - College Algebra

Question

For the following exercises, simplify the rational expression. $$ \frac{\frac{2}{a}+\frac{7}{b}}{b} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2b + 7a}{ab^2} \).

Step 1: Combine the numerators

First, identify the separate fraction terms in the numerator, which are \( \frac{2}{a} + \frac{7}{b} \). To combine these fractions into a single fraction, find a common denominator for \( a \) and \( b \), which is \( ab \). Rewrite each term with the common denominator:\[\frac{2}{a} = \frac{2b}{ab}, \quad \text{and} \quad \frac{7}{b} = \frac{7a}{ab}\]Thus, the combined numerator becomes:\[\frac{2b + 7a}{ab}\]

Step 2: Substitute back into the original expression

Replace the original numerator \( \frac{2}{a} + \frac{7}{b} \) with the simplified form we found in Step 1:\[\frac{\frac{2b + 7a}{ab}}{b}\]

Step 3: Simplify the complex fraction

Understand the expression \( \frac{\frac{2b + 7a}{ab}}{b} \) as dividing \( \frac{2b + 7a}{ab} \) by \( b \). To divide a fraction by a whole number, multiply by its reciprocal:\[\frac{2b + 7a}{ab} \times \frac{1}{b} = \frac{2b + 7a}{ab \cdot b}\]Simplify the denominator to get:\[\frac{2b + 7a}{ab^2}\]

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying these expressions means reducing them to their simplest form.

• Start by identifying common factors in the numerator and denominator.
• Factorize these polynomials if possible to cancel out common terms.

In our given problem, we combined two fractions in the numerator and found a common denominator for them. This common denominator was used to write the sum as a single fraction in the simplest form. Once expressed as a single fraction, you can move to simplify the whole rational expression, often by canceling out common terms in the numerator and the denominator.

Complex Fractions

Complex fractions have a fraction in the numerator, denominator, or both. They can initially look quite intimidating. However, simplifying them can be quite straightforward once you know the steps.

• The first step is to simplify the complex fraction by turning it into a simpler rational expression.
• This involves finding a common denominator for fractions within the numerator or the denominator.

For example, our original expression involved a fraction within another fraction. Simplifying such expressions typically involves converting it to a simpler fraction by multiplying by the reciprocal of the denominator. This reduces the complexity and makes it easier to handle.

Common Denominators

When working with rational expressions that have more than one fraction term, finding a common denominator is essential. This helps in combining fractions into a single expression.

• A common denominator is a shared multiple of the denominators of the fractions involved.
• Finding the least common denominator often helps to simplify expressions faster.

In the exercise, the common denominator for the fractions in the numerator was determined as the product of their individual denominators: \( a \times b \). By rewriting each fraction with this common denominator, it became possible to combine the terms into a single rational expression, which was crucial for the simplification process. Understanding and finding common denominators is an essential skill for simplifying more complicated rational expressions.