Problem 18 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{b^{3}-8}{10 b^{4}}+\frac{8}{10 b^{4}}\)

Step-by-Step Solution

Answer

\(\frac{1}{10b}\) is the simplified result.

Step 1: Identify the common denominator

Both fractions have the same denominator: \(10b^4\). This means they can be combined directly.

Step 2: Combine the numerators

Since the denominators are the same, add the numerators: \(\frac{b^3 - 8}{10b^4} + \frac{8}{10b^4} = \frac{(b^3 - 8) + 8}{10b^4}\).

Step 3: Simplify the numerator

Simplify the expression in the numerator: \((b^3 - 8) + 8 = b^3\). So the fraction becomes \(\frac{b^3}{10b^4}\).

Step 4: Simplify the fraction

Simplify \(\frac{b^3}{10b^4}\) by dividing both the numerator and the denominator by \(b^3\). This gives: \(\frac{1}{10b}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in algebra that helps us reduce fractions to their simplest form. This makes them easier to understand and work with. The simplification process involves reducing the numerator and the denominator so that they no longer share any common factors besides 1. In our problem, we simplified the fraction \(\frac{b^3}{10b^4}\). Here's how we did it:

• Identify common factors in the numerator and the denominator.
• In \(\frac{b^3}{10b^4}\), both the numerator and the denominator share \(b^3\) as a common factor.
• Divide both by \(b^3\), to get our simplified fraction \(\frac{1}{10b}\).

Simplifying fractions helps us recognize equivalent fractions more easily and often makes computations simpler. Remember, when both numerator and denominator are divided by the same value, the value of the fraction remains unchanged.

Common Denominator

Finding a common denominator is an essential step when adding or subtracting fractions. This ensures that the fractions have a uniform basis to be combined. In the exercise we solved, both fractions \(\frac{b^3 - 8}{10b^4}\) and \(\frac{8}{10b^4}\) already had a common denominator, \(10b^4\). Here's why it's important:

• Fractions must share the same denominator to be directly added or subtracted.
• Having a common denominator allows for straightforward combination of the numerators; there's no extra work of adjusting the denominators.
• In this case, recognizing the shared denominator \(10b^4\) streamlined the addition process.
• This consistency in the denominator also directly impacted our ability to simplify the expression later on.

When fractions do not initially share a common denominator, it's important to find the least common denominator to ensure calculations can proceed smoothly.

Numerator and Denominator

Understanding what numerators and denominators represent is key in grasping how fractions work. In a fraction, the numerator is the top number, and represents parts of a whole or a specific value under discussion. The denominator is the bottom number and indicates the total parts into which the whole is divided. Consider this through our exercise:

• In \(\frac{b^3 - 8}{10b^4}\) and \(\frac{8}{10b^4}\), the numerators are \(b^3 - 8\) and \(8\) respectively.
• The denominator \(10b^4\) shows that both expressions are part of the same entire group defined by this term.
• Combining numerators was possible because they shared the same denominator, allowing us to express a single fraction.
• Simplifying these components further clarified the expression to \(\frac{1}{10b}\), cutting out redundancy and making it easier to interpret.

Understanding these basic components of fractions provides the foundation for more complex operations like addition, subtraction, and simplification.