Problem 10 - Elementary Algebra

Question

\(\left(24 c^{3}-8 c\right) \div(8 c)\)

Step-by-Step Solution

Answer

3c^2 - 1

Step 1: Identify the Expression

First, identify the given expression: (24c^3 - 8c) / (8c)

Step 2: Distribute the Denominator

Distribute the denominator, 8c, to each term in the numerator: (24c^3) / (8c) - (8c) / (8c)

Step 3: Simplify Each Term

Simplify each separate fraction: (24c^3) / (8c) = 3c^2 (8c) / (8c) = 1

Step 4: Combine the Results

Combine the simplified terms to get the final answer: 3c^2 - 1

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. This process involves reducing the expression to its simplest form by performing basic arithmetic operations and applying algebraic rules.
To simplify the expression \((24c^3 - 8c) / (8c)\), follow these steps:

• Identify each term in the numerator.
• Distribute the denominator to each term in the numerator separately.
• Reduce the fractions by canceling out common factors.
• Combine the simplified terms to get the final result.

In this example, we separated the original expression into two simpler expressions, \(\frac{24c^3}{8c}\) and \(\frac{8c}{8c}\), and then simplified each independently. This step-by-step approach makes complex expressions more manageable.

Fraction Distribution

Fraction distribution involves spreading a single denominator across multiple terms in the numerator. This makes it easier to simplify each term separately. Here's the process using our example:

• Start with the expression: \((24c^3 - 8c) / (8c)\).
• Distribute the denominator \(8c\) to each term in the numerator: \((24c^3)/(8c) - (8c)/(8c)\).
• By doing this, we create separate fractions that can be simplified individually.

Distributing the denominator helps in breaking down complex fractions into smaller, simpler parts that are easier to handle. For our example, it makes it clear which parts of the expression we need to simplify next.

Algebraic Fractions

Understanding algebraic fractions is key in algebra. These fractions have variables in the numerator, the denominator, or both. Here's a breakdown using our example:

• An algebraic fraction like \((24c^3 - 8c) / (8c)\) involves terms with variables.
• To simplify, we distributed the denominator \(8c\) to each term in the numerator.
• We then simplified each fraction: \((24c^3) / (8c) = 3c^2\) and \(8c / 8c = 1\).

Combining these results, we get \(3c^2 - 1\).
For any algebraic fraction, always distribute the denominator first, simplify each fraction, then combine the simplified results. Practice makes this process much easier to handle, helping you with more complex algebraic problems in the future.