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:

For comparisons of $n^{1/n}$, note that the sequence increases up to $n=3$ and then decreases.

Answer 1

The Correct Option is B

Approximating each value: \[ 2^{1/2} \approx 1.4142 \] \[ 3^{1/3} \approx 1.4422 \] \[ 4^{1/4} \approx 1.4142 \] \[ 6^{1/6} \approx 1.3480 \] \[ 12^{1/12} \approx 1.2311 \] The largest is clearly $3^{1/3} \approx 1.4422$. \[ \boxed{3^{1/3}} \]