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 Expression Inside Parentheses

Begin with the expression inside the parentheses: \( \frac{1}{4} - \frac{1}{3} \). To subtract the fractions, ensure they have a common denominator. The least common denominator (LCD) of 4 and 3 is 12. Rewrite each fraction: \( \frac{1}{4} = \frac{3}{12} \) and \( \frac{1}{3} = \frac{4}{12} \). Now subtract: \( \frac{3}{12} - \frac{4}{12} = -\frac{1}{12} \).

Step 2: Simplify Inside the Brackets

With the result from Step 1, substitute back into the expression: \( \frac{1}{2} \left(-\frac{1}{12}\right) + \frac{1}{6} \). First, calculate \( \frac{1}{2} \times \left(-\frac{1}{12}\right) = -\frac{1}{24} \). Now add \( -\frac{1}{24} + \frac{1}{6} \). Convert \( \frac{1}{6} \) to \( \frac{4}{24} \) (with a common denominator of 24) and perform the addition: \( -\frac{1}{24} + \frac{4}{24} = \frac{3}{24} \). Reduce \( \frac{3}{24} \) to \( \frac{1}{8} \).

Step 3: Compute the Final Expression

Substitute the result from Step 2 into the expression \( \frac{1}{3} \left[\frac{1}{8}\right] \). Multiply: \( \frac{1}{3} \times \frac{1}{8} = \frac{1}{24} \). This is the simplified form of the original expression.

Key Concepts

Common Denominator

In fraction operations, like addition or subtraction, having a common denominator is crucial. It allows us to directly compare the fractions or perform arithmetic operations on them. A common denominator is a shared multiple of the denominators of the fractions involved. To find it, we often look for the smallest shared multiple, known as the Least Common Denominator (LCD).

For instance, in our exercise with the fractions \( \frac{1}{4} \) and \( \frac{1}{3} \), the denominators are 4 and 3. The smallest number that both 4 and 3 divide into evenly is 12. Thus, 12 is the least common denominator. This allows us to rewrite both fractions with this common denominator:

• \( \frac{1}{4} = \frac{3}{12} \)
• \( \frac{1}{3} = \frac{4}{12} \)

This transformation enables the subtraction process to proceed smoothly and clearly.

Fraction Subtraction

Subtracting fractions can be straightforward once you have a common denominator. Once fractions are rewritten to have the same denominator, the numerators can be directly subtracted from one another. This is a key aspect of dealing with fractions in operations.

In our example, after finding the common denominator, we performed the subtraction \( \frac{3}{12} - \frac{4}{12} \). This operation simply involves subtracting the numerators while keeping the common denominator:

• Numerator: \( 3 - 4 = -1 \)
• Denominator remains: 12

The result of this operation is \( -\frac{1}{12} \), which seamlessly integrates into further operations without any additional adjustments to the denominators.

Fraction Multiplication

Fraction multiplication is often simpler than addition or subtraction because no common denominator is required. Instead, we multiply the numerators together and the denominators together.

In the given exercise, we encountered this during the following operation: \( \frac{1}{2} \times -\frac{1}{12} \). The task is straightforward:

• Multiply the numerators: \( 1 \times -1 = -1 \)
• Multiply the denominators: \( 2 \times 12 = 24 \)

Thus, the product is \( -\frac{1}{24} \). Each subsequent multiplication, such as the final step with \( \frac{1}{3} \times \frac{1}{8} \), follows the same method, easily yielding results that maintain simplicity and accuracy. This technique is uniform across all fraction multiplications, making it a reliable skill when handling fractions.