Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(\frac{3}{4-7}+\frac{3}{21}-\frac{1}{6}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{43}{42}\).

Step 1: Simplify Denominators

We start by simplifying the denominator of the first fraction. The expression for the denominator is \(4 - 7\), which simplifies to \(-3\). Thus, the first term becomes \(\frac{3}{-3}\).

Step 2: Simplify the First Fraction

Now, simplify the fraction \(\frac{3}{-3}\). Both the numerator and the denominator can be divided by 3, which gives us \(-1\).

Step 3: Common Denominator for Addition

Next, let's simplify \(\frac{3}{21}\) and \(\frac{1}{6}\) to find a common denominator. \(\frac{3}{21}\) simplifies to \(\frac{1}{7}\) by dividing the numerator and denominator by 3.

Step 4: Determine Common Denominator

We now need the least common denominator (LCD) for \(\frac{1}{7}\) and \(\frac{1}{6}\), which is 42. Convert each fraction: \(\frac{1}{7} = \frac{6}{42}\) and \(\frac{1}{6} = \frac{7}{42}\).

Step 5: Rewrite Original Expression with Common Denominator

Rewrite the entire expression using the simplified fractions from Step 2 and Step 4: \(-1 + \frac{6}{42} - \frac{7}{42}\).

Step 6: Solve Expression with Common Denominators

Combine the fractions with common denominator: \(\frac{6}{42} - \frac{7}{42} = \frac{6 - 7}{42} = \frac{-1}{42}\).

Step 7: Combine All Simplified Parts

Finally, add \(-1\) and \(\frac{-1}{42}\): \(-1 + \frac{-1}{42} = -1 - \frac{1}{42}\).

Step 8: Final Simplified Result

Convert the expression to a single fraction: \(-1 = -\frac{42}{42}\), then combine \(-\frac{42}{42} - \frac{1}{42} = -\frac{43}{42}\).

Key Concepts

Fraction Simplification

When simplifying a fraction, the goal is to make it as simple as possible by dividing both the numerator and the denominator by their greatest common factor. So, let’s explore this concept further.
To simplify a fraction, follow these basic steps:

• Identify the numerator (the number above the line).
• Identify the denominator (the number below the line).
• Find the greatest common divisor (GCD) for both numbers.
• Divide both the numerator and denominator by this GCD.

For example, if we take the fraction \(\frac{3}{9}\), the GCD of 3 and 9 is 3, so dividing both by 3 simplifies it to \(\frac{1}{3}\).
Simplification makes fractions easier to work with by reducing their size, thus making calculations more manageable. It is crucial for combining different fractions, like those in our problem.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. A common denominator is like a common language for fractions, which allows them to communicate and work together.The steps to find a common denominator are:

• Find the least common multiple (LCM) of the denominators you have. This becomes your common denominator.
• Adjust each fraction so that its denominator is equal to the common denominator by multiplying both the numerator and denominator by the necessary factors.

In our problem, the fractions are \(\frac{1}{7}\) and \(\frac{1}{6}\). Their common denominator is 42. Here's how we adjust them:
- Convert \(\frac{1}{7}\) to \(\frac{6}{42}\) by multiplying the numerator and denominator by 6.- Convert \(\frac{1}{6}\) to \(\frac{7}{42}\) by multiplying the numerator and denominator by 7.Once both fractions share a common denominator, you can add or subtract them by performing these operations on the numerators, keeping the common denominator the same.

Step-by-Step Solution

Breaking down math problems into smaller, digestible steps is an effective strategy for problem-solving. Let's see how this approach works in simplifying fractions.First, tackle any complex parts of the problem, such as simplifying the expressions in this exercise: - Begin by simplifying the fraction's denominator: \(4 - 7\) becomes \(-3\), simplifying \(\frac{3}{-3}\) to \(-1\).Next, focus on finding common ground for operations, like converting fractions to a common denominator:- Simplify \(\frac{3}{21}\) to \(\frac{1}{7}\) and determine the LCD for \(\frac{1}{7}\) and \(\frac{1}{6}\), which is 42.Then rewrite the expression using this common denominator:- Transform \(-1 + \frac{6}{42} - \frac{7}{42}\) and take care of subtraction: \(\frac{6}{42} - \frac{7}{42} = \frac{-1}{42}\).Finally, combine all results for the solution:- Merge it into a single simplified result, like combining \(-1 = -\frac{42}{42}\) with \(\frac{-1}{42}\), resulting in \(-\frac{43}{42}\).Using step-by-step methods makes complex problems more approachable, ensuring no detail is overlooked while boosting comprehension.