Question

Which of the terms \( 2^{1/3}, 3^{1/4}, 4^{1/6}, 6^{1/8}, 10^{1/12} \) is the largest?

• A. \( 2^{1/3} \)
• B. \( 3^{1/4} \)
• C. \( 4^{1/6} \)
• D. \( 10^{1/12} \)

Hint:

Use logarithms to compare exponential expressions with fractional powers.

Answer 1

The Correct Option is B

To compare terms like \( a^{1/n} \), use logarithms:
Let \( y = a^{1/n} \Rightarrow \log y = \frac{1}{n} \log a \).
Compute values: \[ \log(2^{1/3}) = \frac{1}{3} \log 2 \approx 0.1003
\log(3^{1/4}) = \frac{1}{4} \log 3 \approx 0.119
\log(4^{1/6}) = \frac{1}{6} \log 4 \approx 0.1002
\log(6^{1/8}) = \frac{1}{8} \log 6 \approx 0.096
\log(10^{1/12}) = \frac{1}{12} \log 10 = 0.0833 \] So the maximum log value is for option (B). Hence, the largest value is: \[ \boxed{3^{1/4}} \]