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Chapter 4: Problem 33

Evaluate each expression. $$ \left(\frac{1}{2}\right)^{-1}+\left(\frac{2}{3}\right)^{-1} $$

Short Answer

Expert verified
\(\frac{7}{2}\)

Step by step solution

01

Understanding the Problem

The problem requires evaluating the expression \(\left(\frac{1}{2}\right)^{-1}+\left(\frac{2}{3}\right)^{-1}\). This involves finding the reciprocal of each fraction and then adding the results.
02

Find the Reciprocal of \(\frac{1}{2}\)

To find the reciprocal of \(\frac{1}{2}\), we take the value and flip the numerator and denominator. Therefore, the reciprocal of \(\frac{1}{2}\) is \(2\).
03

Find the Reciprocal of \(\frac{2}{3}\)

Similarly, to find the reciprocal of \(\frac{2}{3}\), we flip the numerator and denominator. Thus, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
04

Add the Reciprocals

Now, we add the reciprocals obtained: \(2 + \frac{3}{2}\). To add these fractions, first convert the integer \(2\) to a fraction: \(2 = \frac{4}{2}\). Now, add the fractions: \(\frac{4}{2} + \frac{3}{2} = \frac{7}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocals
To evaluate the given expression, you first need to understand the concept of reciprocals. A reciprocal of a number is simply flipping the numerator and the denominator of a fraction. For example, the reciprocal of \(\frac{1}{2}\) is \(2\), and the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
Reciprocals are often used in division and are a fundamental part of understanding how fractions work. When you multiply a number by its reciprocal, the result is always one, e.g., \(\frac{1}{2} \times 2 = 1\). This is because each term cancels the other out completely.
Fraction Addition
The next step in evaluating the expression is adding the results of the reciprocals. Adding fractions involves a few key steps.
First, ensure that both fractions have a common denominator. In our case, given \(2 \text{ or } \frac{4}{2}\) and \(\frac{3}{2}\), both already have the same denominator, 2. This step is crucial as fractions MUST share a common denominator to be added directly.
  • Convert any integer into a fraction if needed, i.e., converting 2 into \(\frac{4}{2}\).
  • After conversion, simply add the numerators because the denominators are the same.
Thus, adding results: \(\frac{4}{2} + \frac{3}{2} = \frac{7}{2}\).
Simplifying Fractions
After obtaining the sum \(\frac{7}{2}\), you should simplify the fraction if possible. Although in this case, \(\frac{7}{2}\) is already in its simplest form, it's a valuable skill to check and simplify fractions as follows:
1. **Identify the Greatest Common Divisor (GCD):** For \(\frac{7}{2}\), identify any common factors between 7 and 2. Here, the GCD is 1 because 7 and 2 do not share any other divisors.
2. **Divide the numerator and the denominator by the GCD:** In this example, since the GCD is 1, dividing by 1 will not change the fraction, so \(\frac{7}{2}\) remains the same.
  • This step ensures the fraction is in its simplest form.
  • Always verify the result by checking if both numerator and denominator are as reduced as possible.
Once simplified, your final answer \( \frac{7}{2} \) is the evaluated expression.

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Most popular questions from this chapter

Use scientific notation to calculate the answer to each problem. See Examples 3-5. During the \(2007-2008\) season, Broadway shows grossed a total of \(9.38 \times 10^{8}\) dollars. Total attendance for the season was \(1.23 \times 10^{7} .\) What was the average ticket price for a Broadway show? (Source: The Broadway League.)

Find each product. $$ \left(2 x^{2}-5\right)\left(2 x^{2}+5\right) $$

Each statement contains a number in boldface italics. Write the number in scientific notation. In \(2007,\) the leading U.S. advertiser was the Procter and Gamble Company, which spent approximately 5,230,000,000 dollars. (Source: Crain Communications, Inc.)

Find each product. $$ -3 k\left(8 k^{2}-12 k+2\right) $$

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 2 x^{2}-3 x+10 $$
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