The least number among the numbers √2, 41/3, 51/4, 31/6 is:

Concept: To compare numbers written as different roots (fractional exponents with different denominators), rewrite every exponent with the same denominator — the least common multiple (LCM) of the denominators. Once every value carries an identical exponent, comparing them reduces to comparing their bases: for a common positive exponent, a larger base gives a larger value and a smaller base gives a smaller value.

Application: The four given numbers, written as fractional exponents, are 21/2, 41/3, 51/4, and 31/6.

1. The exponent denominators are 2, 3, 4 and 6; their LCM is 12.

2. Rewrite each exponent over the common denominator 12: 26/12, 44/12, 53/12, and 32/12.

3. Convert each to a twelfth root of a single base: 26/12 = (26)1/12 = 641/12; 44/12 = (44)1/12 = 2561/12; 53/12 = (53)1/12 = 1251/12; 32/12 = (32)1/12 = 91/12.

4. With every value now expressed as a twelfth root, compare the bases 64, 256, 125 and 9: the smallest base is 9, so 91/12, i.e. 31/6, is the least of the four numbers.

Cross-check: Approximating each value confirms the ordering: 21/2 ≈ 1.414, 41/3 ≈ 1.587, 51/4 ≈ 1.495, and 31/6 ≈ 1.201 — the smallest decimal value, matching the base comparison above.