Problem 64 - Algebra for College Students

Question

Use a calculator to evaluate each complex fraction. Round answers to four decimal places. If your calculator does fractions, then also find the fractional answer. $$\frac{2^{-1}+3^{-1}-6^{-1}}{3^{-1}-5^{-1}+4^{-1}}$$

Step-by-Step Solution

Answer

The fractional answer is \(\frac{40}{23}\) and the decimal answer is approximately 1.7391.

Step 1 - Express Negative Exponents as Fractions

Recall that a negative exponent means the reciprocal of the base raised to the positive exponent. Therefore, express each term with a negative exponent as a fraction: \(2^{-1} = \frac{1}{2}\), \(3^{-1} = \frac{1}{3}\), \(6^{-1} = \frac{1}{6}\), \(5^{-1} = \frac{1}{5}\), and \(4^{-1} = \frac{1}{4}\).

Step 2 - Substitute Terms into Expression

Replace the negative exponents in the original expression with their fractional equivalents: \[\frac{\frac{1}{2} + \frac{1}{3} - \frac{1}{6}}{\frac{1}{3} - \frac{1}{5} + \frac{1}{4}}\]

Step 3 - Find Common Denominators and Simplify Numerator

To add and subtract fractions, find a common denominator for the fractions in the numerator: The common denominator for \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\) is 6: \[\frac{1}{2} = \frac{3}{6}\, , \frac{1}{3} = \frac{2}{6}\, , \frac{1}{6} = \frac{1}{6}\]. Combine these to get: \[\frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\]

Step 4 - Find Common Denominators and Simplify Denominator

Next, find a common denominator for the fractions in the denominator: The common denominator for \(\frac{1}{3}\), \(\frac{1}{5}\), and \(\frac{1}{4}\) is 60: \[\frac{1}{3} = \frac{20}{60}\, , \frac{1}{5} = \frac{12}{60}\, , \frac{1}{4} = \frac{15}{60}\]. Combine these to get: \[\frac{20}{60} - \frac{12}{60} + \frac{15}{60} = \frac{23}{60}\]

Step 5 - Simplify The Entire Expression

Now, divide the simplified numerator by the simplified denominator: \[\frac{\frac{2}{3}}{\frac{23}{60}} = \frac{2}{3} \times \frac{60}{23} = \frac{120}{69} = \frac{40}{23}\]. To get the decimal answer, perform the division: \[\frac{40}{23} \approx 1.7391\]

Key Concepts

Negative Exponents

Understanding negative exponents is key to solving the problem given. A negative exponent indicates that you take the reciprocal of the base and then raise it to the positive exponent. For example, if you have a term like \(x^{-n}\), you can rewrite it as \(\frac{1}{x^n}\).
Let's break down the negative exponents in the exercise:

• \(2^{-1}\) becomes \(\frac{1}{2}\)
• \(3^{-1}\) becomes \(\frac{1}{3}\)
• \(6^{-1}\) becomes \(\frac{1}{6}\)
• \(5^{-1}\) becomes \(\frac{1}{5}\)
• \(4^{-1}\) becomes \(\frac{1}{4}\)

Substituting these into the original expression makes it easier to handle fractions.

Fraction Simplification

Simplifying fractions is essential for combining terms correctly. When dealing with fractions, the key is to find a common denominator so you can add or subtract the fractions properly. This involves:

• Converting each term to have the same denominator.
• Performing addition or subtraction as needed.

In the exercise:
For the numerator, we have \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\). The common denominator is 6:

• \(\frac{1}{2} = \frac{3}{6}\)
• \(\frac{1}{3} = \frac{2}{6}\)
• \(\frac{1}{6} = \frac{1}{6}\)

Combining these, we get:
\(\frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\)
For the denominator, we have \(\frac{1}{3}\), \(\frac{1}{5}\), and \(\frac{1}{4}\). The common denominator is 60:

• \(\frac{1}{3} = \frac{20}{60}\)
• \(\frac{1}{5} = \frac{12}{60}\)
• \(\frac{1}{4} = \frac{15}{60}\)

Combining these, we get:
\(\frac{20}{60} - \frac{12}{60} + \frac{15}{60} = \frac{23}{60}\)

Common Denominators

Finding common denominators is crucial in operations involving fractions. A common denominator is a shared multiple of the denominators of the fractions you're working with, allowing you to combine the fractions correctly.
In the exercise, to operate on fractions like \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\), we first find the least common multiple (LCM) of 2, 3, and 6, which is 6. This enables us to rewrite each fraction with the same denominator of 6.
Similarly, for \(\frac{1}{3}\), \(\frac{1}{5}\), and \(\frac{1}{4}\), the LCM is 60. Transforming each fraction so they all have a denominator of 60 facilitates addition or subtraction.
By using common denominators:

• The numerator becomes \(\frac{2}{3}\) after combining.
• The denominator becomes \(\frac{23}{60}\) after combining.

These simplifications allow you to carry out division between the combined numerator and denominator effectively, leading to the final simplified form and a correct decimal approximation.