Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.$$ \frac{1}{3}\left[\frac{1}{2}\left(\frac{1}{4}-\frac{1}{3}\right)+\frac{1}{6}\right] $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{24}\).

Step 1: Simplify the Inner Parentheses

Start with the expression inside the parentheses: \[\frac{1}{4} - \frac{1}{3}\]Find a common denominator, which is 12, for the fractions and simplify:\[\frac{3}{12} - \frac{4}{12} = -\frac{1}{12}\]

Step 2: Multiply by the Inner Coefficient

Take the result from Step 1 and multiply it by \(\frac{1}{2}\):\[\frac{1}{2} \times -\frac{1}{12} = -\frac{1}{24}\]

Step 3: Add the Second Fraction Inside the Brackets

Add the result from Step 2 to the fractional term \(\frac{1}{6}\):\[-\frac{1}{24} + \frac{1}{6}\]Convert \(\frac{1}{6}\) to \(\frac{4}{24}\) to have a common denominator:\[-\frac{1}{24} + \frac{4}{24} = \frac{3}{24}\]Simplify \(\frac{3}{24}\) to \(\frac{1}{8}\).

Step 4: Multiply by the Outer Coefficient

Multiply the result from Step 3 by \(\frac{1}{3}\):\[\frac{1}{3} \times \frac{1}{8} = \frac{1}{24}\]

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics, involving reducing expressions to their most basic form. It requires basic arithmetic operations such as addition, subtraction, multiplication, and division.

In our exercise, our main goal was to simplify a complex fraction by reducing it step by step.

• First, focus on the inner expression, which involves subtraction of fractions.
• Handle the operations inside the parentheses first to simplify parts of the expression separately.
• Systematically remove parentheses and perform operations in an orderly manner, one step at a time.

After simplification, make sure no fractions or numbers inside are more complex than necessary. This ensures the expression is as simplified as possible, making it easier to work with or interpret.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. A common denominator is a shared multiple of the denominators of the fractions involved.

To operate on fractions:

• Determine the least common denominator (LCD) between the fractions. This is typically the smallest number into which both denominators can divide completely.
• Convert the fractions to equivalent fractions that have the LCD as their denominator. Adjust the numerators accordingly to ensure the fractions remain equivalent.
• Once both fractions share the same denominator, you can easily add or subtract the numerators while keeping the denominator constant.

In our solution, the common denominator for \(\frac{1}{4} and \frac{1}{3}\) was 12, which allowed us to subtract the two fractions effectively. This step is vital for ensuring accuracy in dealing with fractional calculations.

Algebraic Manipulation

Algebraic manipulation involves working with algebraic expressions using various mathematical rules and techniques to simplify or solve them.

In our example, we applied algebraic manipulation in several steps:

• Firstly, subtraction and simplification of the fractions.
• Next, multiplication of fractions by a coefficient to scale or adjust them.
• Finally, using equivalent fractions for addition or subtraction when necessary.

This operation enhances our ability to deal with expressions, especially in equations or complex algebraic problems. By consistently applying these rules and strategies, we can transform challenging algebraic tasks into manageable steps that lead us to the correct solution.