Convert radicals to fractional exponents

Rewrite $\sqrt{2}$ as $2^{1/2}$, $3^{1/3}$ remains as is, and $4^{1/4}$ can be simplified if desired, but we will proceed with the original values.

$2^{1/2}, 3^{1/3}, 4^{1/4}$

Find the LCM of denominators

The denominators are 2, 3, and 4. Their LCM is 12.

$\text{LCM}(2, 3, 4) = 12$

Raise each term to the power of 12

Calculate $(2^{1/2})^{12} = 2^6 = 64$, $(3^{1/3})^{12} = 3^4 = 81$, and $(4^{1/4})^{12} = 4^3 = 64$.

$2^6 = 64, 3^4 = 81, 4^3 = 64$

Compare results

Comparing the values 64, 81, and 64, clearly 81 is the greatest.

$81 > 64$