To evaluate the expression \(-3 * 4\) using the operation defined by \(m * n = \left(\frac{m}{n} - \frac{n}{m}\right)\), we will follow these steps: ### Step 1: Substitute the values of \(m\) and \(n\) Here, we have: - \(m = -3\) - \(n = 4\) Now, we substitute these values into the operation definition: \[ -3 * 4 = \left(\frac{-3}{4} - \frac{4}{-3}\right) \] ### Step 2: Calculate each fraction 1. **Calculate \(\frac{-3}{4}\)**: \[ \frac{-3}{4} = -0.75 \] 2. **Calculate \(\frac{4}{-3}\)**: \[ \frac{4}{-3} = -\frac{4}{3} \approx -1.3333 \] ### Step 3: Combine the fractions Now we need to combine these two fractions: \[ -3 * 4 = \left(\frac{-3}{4} - \left(-\frac{4}{3}\right)\right) = \left(\frac{-3}{4} + \frac{4}{3}\right) \] ### Step 4: Find a common denominator To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. We will convert both fractions: 1. **Convert \(\frac{-3}{4}\)** to have a denominator of 12: \[ \frac{-3}{4} = \frac{-3 \times 3}{4 \times 3} = \frac{-9}{12} \] 2. **Convert \(\frac{4}{3}\)** to have a denominator of 12: \[ \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \] ### Step 5: Add the fractions Now we can add the two fractions: \[ -3 * 4 = \frac{-9}{12} + \frac{16}{12} = \frac{-9 + 16}{12} = \frac{7}{12} \] ### Final Answer Thus, the value of \(-3 * 4\) is: \[ \frac{7}{12} \] ### Explanation of Options Now, let's analyze the options provided: - **A. \(\frac{-25}{12}\)**: This option is incorrect because it does not match our calculated result. It suggests a negative value that does not arise from our operation. - **B. \(\frac{-7}{12}\)**: This option is incorrect. It is the negative of our correct answer, which indicates a misunderstanding of the operation's result. - **C. \(\frac{7}{12}\)**: This is the correct answer, as we calculated above. - **D. \(\frac{25}{12}\)**: This option is incorrect as it suggests a positive value that is not relevant to our calculation. ### Common Pitfalls 1. **Misunderstanding the operation**: Ensure you correctly apply the operation defined in the question. 2. **Fraction addition**: Remember to find a common denominator when adding fractions. 3. **Sign errors**: Be careful with negative signs, especially when subtracting or adding negative fractions. ### Revision Summary - The operation \(m * n\) is defined as \(\left(\frac{m}{n} - \frac{n}{m}\right)\). - Substitute values carefully and calculate each fraction. - Use a common denominator to add fractions correctly. - The final answer for \(-3 * 4\) is \(\frac{7}{12}\).