Problem 33 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{4}{5 b}+\frac{1}{b-1}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{9b - 4}{5b(b - 1)}\).

Step 1: Identify the Least Common Denominator (LCD)

The denominators in the expression are \(5b\) and \(b - 1\). To add these fractions, we need a common denominator. The least common denominator is \(5b(b - 1)\).

Step 2: Rewrite Each Fraction with the LCD

Transform each fraction so that they both have the least common denominator. Multiply the numerator and denominator of the first fraction \(\frac{4}{5b}\) by \(b - 1\), giving \(\frac{4(b - 1)}{5b(b - 1)}\). Multiply the numerator and denominator of the second fraction \(\frac{1}{b-1}\) by \(5b\), giving \(\frac{5b}{5b(b-1)}\).

Step 3: Add the Fractions

Now that the fractions have the same denominator, add the numerators: \(4(b - 1) + 5b\). The common denominator is \(5b(b - 1)\), so we have \(\frac{4(b - 1) + 5b}{5b(b - 1)}\).

Step 4: Simplify the Numerator

Expand and simplify the expression in the numerator: \(4(b - 1) + 5b = 4b - 4 + 5b = 9b - 4\). This gives the fraction as \(\frac{9b - 4}{5b(b - 1)}\).

Step 5: Review for Further Simplification

Check whether \(9b - 4\) can be factored further to simplify the fraction. Since \(9b - 4\) doesn't factor, the expression \(\frac{9b - 4}{5b(b - 1)}\) is in its simplest form.

Key Concepts

Least Common Denominator

When working with rational expressions, finding the least common denominator (LCD) is vital for adding or subtracting fractions. The LCD enables the expression to have a common base, making it easier to combine fractions. In the expression \( \frac{4}{5b} + \frac{1}{b-1} \), the denominators are \( 5b \) and \( b-1 \). To compute their LCD, we must multiply these denominators, since they cannot be easily combined or simplified otherwise. Thus, the LCD becomes \( 5b(b - 1) \). This technique ensures that the fractions can be rewritten with a common denominator, which is necessary before performing any addition or subtraction.

Fraction Addition

Once the fractions have the same denominator, they can be easily added. The process is simple: add the numerators while keeping the denominator the same. In our task, after rewriting \( \frac{4}{5b} \) as \( \frac{4(b - 1)}{5b(b - 1)} \) and \( \frac{1}{b - 1} \) as \( \frac{5b}{5b(b - 1)} \), the fractions are ready to be combined. To perform the addition, simply add the numerators: \( 4(b - 1) + 5b \). The denominator remains \( 5b(b - 1) \). This step is straightforward, due to the preparation done in finding and applying the LCD.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. After adding the fractions, you must simplify the numerator, if possible. For the given case, expanding \( 4(b - 1) + 5b \) results in \( 4b - 4 + 5b \), which simplifies further to \( 9b - 4 \). Once simplified, the expression is written as \( \frac{9b - 4}{5b(b - 1)} \). During this step, it's crucial to check whether the numerator or denominator can be factored further. If not, as seen here with \( 9b - 4 \), the expression is already in its simplest form. Simplification ensures that your answer is as concise as possible, which is important for clarity and correctness.

Algebraic Operations

Algebraic operations in rational expressions involve several key skills: identifying the LCD, manipulating numerators and denominators, and combining and simplifying fractions. Each step requires a careful approach.

• Identifying the LCD ensures commonality in the expression's base.
• Rewriting fractions with this common base simplifies their combination.
• Adding the numerators allows the overall simplification of the expression.
• Simplifying the expression ensures its clarity and correctness.

By following these steps methodically, you can master the manipulation of rational expressions, ensuring each step is clear and understandable, thus building a strong foundation in algebra.