Question

Which among \( 2^{1/2}, 3^{1/3}, 4^{1/4}, 6^{1/6} \) and \( 12^{1/12} \) is the largest?

• A. \( 2^{1/2} \)
• B. \( 3^{1/3} \)
• C. \( 4^{1/4} \)
• D. \( 6^{1/6} \)
• E. \( 12^{1/12} \)

Hint:

The function \( f(x) = x^{1/x} \) increases until \( x = e \approx 2.718 \) and then decreases. Since \( 3 \) is the integer closest to \( e \), \( 3^{1/3} \) is the largest among such terms.

Answer 1

The Correct Option is B

Step 1: Understanding the problem:
We are asked to find the largest value among the following expressions: \[ 2^{1/2}, \quad 3^{1/3}, \quad 4^{1/4}, \quad 6^{1/6}, \quad 12^{1/12} \] Step 2: Taking logarithms to compare the values:
Since these are fractional powers, we can take the natural logarithm (ln) of each term to make comparison easier. We will compare the values of the logarithms to determine the largest. We will use the property that \( \ln(a^b) = b \ln(a) \). \[ \ln(2^{1/2}) = \frac{1}{2} \ln(2) \approx \frac{1}{2} \times 0.6931 = 0.3466 \] \[ \ln(3^{1/3}) = \frac{1}{3} \ln(3) \approx \frac{1}{3} \times 1.0986 = 0.3662 \] \[ \ln(4^{1/4}) = \frac{1}{4} \ln(4) = \frac{1}{4} \times 1.3863 = 0.3466 \] \[ \ln(6^{1/6}) = \frac{1}{6} \ln(6) \approx \frac{1}{6} \times 1.7918 = 0.2986 \] \[ \ln(12^{1/12}) = \frac{1}{12} \ln(12) \approx \frac{1}{12} \times 2.4849 = 0.2071 \] Step 3: Comparing the results:
Now, let's compare the calculated logarithms: \[ \ln(2^{1/2}) \approx 0.3466, \quad \ln(3^{1/3}) \approx 0.3662, \quad \ln(4^{1/4}) \approx 0.3466, \quad \ln(6^{1/6}) \approx 0.2986, \quad \ln(12^{1/12}) \approx 0.2071 \] Step 4: Conclusion:
Since \( 3^{1/3} \) has the largest logarithmic value (0.3662), it is the largest among the given expressions. Therefore, the correct answer is \( \boxed{(b)} \).