Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(-\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}\right)\right]\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{9}\).

Step 1: Simplify Inside the Innermost Parentheses

The expression given is \[-\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}\right)\right]\]. Start by simplifying inside the innermost parentheses: \(\frac{1}{3} - \frac{1}{5}\). To do this, find a common denominator, which is 15. Rewriting both fractions,\[\frac{1}{3} = \frac{5}{15}\quad \text{and} \quad\frac{1}{5} = \frac{3}{15}.\] Now subtract:\[\frac{5}{15} - \frac{3}{15} = \frac{2}{15}.\] So, the expression inside the innermost parentheses is \(\frac{2}{15}\).

Step 2: Simplify Middle Expression

Now substitute \(\frac{2}{15}\) back into the expression:\[-\frac{1}{3}\left[\frac{2}{5}-\frac{1}{2}\left(\frac{2}{15}\right)\right].\]Simplify \(\frac{1}{2}\left(\frac{2}{15}\right)\): \[\frac{1}{2} \times \frac{2}{15} = \frac{2}{30} = \frac{1}{15}.\]Substitute back:\[-\frac{1}{3}\left[\frac{2}{5} - \frac{1}{15}\right].\]

Step 3: Simplify Expression in Brackets

Continue by finding a common denominator for \(\frac{2}{5}\) and \(\frac{1}{15}\), which is 15. Rewriting \(\frac{2}{5}\):\[\frac{2}{5} = \frac{6}{15}.\]Now subtract:\[\frac{6}{15} - \frac{1}{15} = \frac{5}{15} = \frac{1}{3}.\]This simplifies the bracket expression to \(\frac{1}{3}\).

Step 4: Multiply with the Outer Factor

Substituting back into the main expression:\[-\frac{1}{3} \left( \frac{1}{3} \right).\]Simplify the multiplication:\[-\frac{1}{3} \times \frac{1}{3} = -\frac{1}{9}.\]

Step 5: Review the Simplification

Verify that all expressions are fully simplified, all parentheses have been removed, and all fractions are reduced. The final simplified result of the original expression is:\(-\frac{1}{9}.\)

Key Concepts

Fraction Reduction

Fraction reduction is the process of simplifying a fraction to its lowest terms. It involves breaking down the numerator and the denominator to their smallest possible values. For example,

• Consider the fraction \( \frac{2}{30} \). Both 2 and 30 can be divided by 2 to obtain \( \frac{1}{15} \).
• Similarly, the fraction \( \frac{5}{15} \) can be simplified by dividing both the top (numerator) and bottom (denominator) by 5 to achieve \( \frac{1}{3} \).

The aim is to simplify fractions whenever possible to make them easier to work with, especially in larger calculations or expressions. It is always ideal in mathematics to present your final answer in its simplest form. Remember that a simplified fraction should have no common factors other than 1 in the numerator and denominator.

Common Denominator

When adding or subtracting fractions, it's necessary to have a common denominator. This involves adjusting each fraction to have the same denominator, allowing for straightforward arithmetic operations.
To find a common denominator:

• Identify the denominators of the fractions involved. For \( \frac{1}{3} \, \text{and}\, \frac{1}{5} \), the denominators are 3 and 5 respectively.
• Determine the smallest number that both denominators can divide into evenly, known as the least common denominator (LCD). Here, the LCD of 3 and 5 is 15.

Once the fractions have the same denominator, their numerators can be easily added or subtracted.
For instance, in the expression \( \frac{1}{3} - \frac{1}{5} \) the fractions are rewritten as \( \frac{5}{15} \text{and} \frac{3}{15} \) before performing the subtraction.

Parentheses Removal

Removing parentheses is an important step when simplifying expressions as it allows you to directly work with and combine terms outside of the nested structure. You should always perform operations inside the parentheses first, then move outward.
Here’s what you need to do:

• Begin with the innermost parentheses, solving them using basic arithmetic or fraction operations.
• Once the expression inside the parentheses is simplified, remove the parentheses and replace them with their simplified form.
• Repeat this process outward until all parentheses are gone.

In our example, simplifying \( \frac{1}{2} \left( \frac{1}{3} - \frac{1}{5} \right) \) gives \( \frac{1}{2} \times \frac{2}{15} = \frac{1}{15} \), allowing us to remove the inner parentheses in a step by step manner.

Arithmetic Operations

Arithmetic operations involve the basic rules of addition, subtraction, multiplication, and division applied to numbers or algebraic expressions. These operations require a clear understanding and careful execution, especially in the context of fractions.

• **Addition and Subtraction**: Fractions must have a common denominator as previously mentioned to proceed with addition or subtraction.
  Example: \( \frac{6}{15} - \frac{1}{15} \) results in \( \frac{5}{15} \) which simplifies to \( \frac{1}{3} \).
• **Multiplication**: Increases or decreases values by multiplying numerators and denominators of the fractions involved.
  Example: \( -\frac{1}{3} \times \frac{1}{3} = -\frac{1}{9} \).

Each step must ensure the correct arithmetic operation is carried out, and fractions are simplified wherever applicable, to ensure the clarity and accuracy of the solution.