To evaluate the expression \((343)^{\frac{1}{3}} \times (0.14)^{-1} \times (25)^{-\frac{1}{2}}\), we will break it down step by step. ### Step 1: Evaluate \((343)^{\frac{1}{3}}\) The expression \((343)^{\frac{1}{3}}\) represents the cube root of 343. We need to find a number that, when multiplied by itself three times, equals 343. - **Finding the cube root**: - We can factor 343. It can be expressed as \(7 \times 7 \times 7\) or \(7^3\). - Therefore, \((343)^{\frac{1}{3}} = (7^3)^{\frac{1}{3}} = 7\). ### Step 2: Evaluate \((0.14)^{-1}\) The expression \((0.14)^{-1}\) is the reciprocal of 0.14. - **Finding the reciprocal**: - The reciprocal of a number \(x\) is \(\frac{1}{x}\). - Thus, \((0.14)^{-1} = \frac{1}{0.14}\). - To convert 0.14 to a fraction, we can write it as \(\frac{14}{100} = \frac{7}{50}\). - Therefore, \((0.14)^{-1} = \frac{50}{7}\). ### Step 3: Evaluate \((25)^{-\frac{1}{2}}\) The expression \((25)^{-\frac{1}{2}}\) represents the negative square root of 25. - **Finding the square root**: - The square root of 25 is 5, since \(5 \times 5 = 25\). - Therefore, \((25)^{-\frac{1}{2}} = \frac{1}{(25)^{\frac{1}{2}}} = \frac{1}{5}\). ### Step 4: Combine all parts Now we can combine all the evaluated parts: \[ (343)^{\frac{1}{3}} \times (0.14)^{-1} \times (25)^{-\frac{1}{2}} = 7 \times \frac{50}{7} \times \frac{1}{5} \] ### Step 5: Simplify the expression 1. **Multiply \(7\) and \(\frac{50}{7}\)**: - The \(7\) in the numerator and denominator cancels out: \[ 7 \times \frac{50}{7} = 50 \] 2. **Now multiply by \(\frac{1}{5}\)**: \[ 50 \times \frac{1}{5} = \frac{50}{5} = 10 \] ### Final Answer Thus, the final answer is \(10\). ### Explanation of Options - **Option A: 10** - This is the correct answer as we have calculated. - **Option B: 12** - This is incorrect because it does not match our calculations. - **Option C: 8** - This is also incorrect; it does not correspond to any part of our evaluation. - **Option D: 7** - This is incorrect as it only represents the cube root of 343, not the entire expression. ### Revision Summary - The cube root of 343 is \(7\). - The reciprocal of \(0.14\) is \(\frac{50}{7}\). - The negative square root of \(25\) is \(\frac{1}{5}\). - Combining these gives \(10\) as the final answer.