Which of the following statements correctly compares the values (65)^(1/6), (17)^(1/4) and (12)^(1/3)?

Introduction / Context:
Comparing numbers expressed as fractional powers, such as sixth roots, fourth roots and cube roots, is a classic aptitude challenge. Direct computation can be messy, so you need to use properties of exponents and logarithms or clever estimation. In this question, three statements propose different orderings of (65)^(1/6), (17)^(1/4) and (12)^(1/3), and you must determine which ordering is correct.

Given Data / Assumptions:

• The three numbers are A = (65)^(1/6), B = (17)^(1/4) and C = (12)^(1/3).
• Statement I: (65)^(1/6) > (17)^(1/4) > (12)^(1/3).
• Statement II: (17)^(1/4) > (65)^(1/6) > (12)^(1/3).
• Statement III: (12)^(1/3) > (17)^(1/4) > (65)^(1/6).
• We assume all roots are real and positive.

Concept / Approach:
To compare numbers like n^(1/k), it is often easier to compare their logarithms or to raise them to a common power. For example, comparing A and B involves comparing 65^(1/6) and 17^(1/4). Instead of finding exact values, we can compare 6th powers or 12th powers, or approximate using basic calculator free reasoning. Another strategy is to note that for fixed exponent 1/k, the function n^(1/k) increases with n, and for fixed base n, the function n^(1/k) decreases as k grows. These ideas help build approximate numerical values that can be ordered.

Step-by-Step Solution:
Step 1: Estimate C = (12)^(1/3). Since 2^3 = 8 and 3^3 = 27, 12 is between 8 and 27, so the cube root of 12 is between 2 and 3. More precisely, it is a little above 2, around 2.28. Step 2: Estimate B = (17)^(1/4). Note that 2^4 = 16 and 3^4 = 81, so the fourth root of 17 is just above 2, maybe around 2.03. Step 3: Estimate A = (65)^(1/6). Since 2^6 = 64 and 3^6 = 729, 65 is just above 64, so the sixth root of 65 is just above 2, but only slightly above, around 2.005. Step 4: From these estimates, we can see that C is largest (about 2.28), B is in the middle (about 2.03) and A is smallest (about 2.01). Step 5: Therefore, the correct inequality is (12)^(1/3) > (17)^(1/4) > (65)^(1/6), which matches statement III.

Verification / Alternative check:
As a more formal alternative without decimals, you can compare squares or higher powers. For instance, to compare B and A, compare 17^(1/4) and 65^(1/6) by raising both to the 12th power, which is the least common multiple of 4 and 6. Then B^12 = 17^3 and A^12 = 65^2. Compute 17^3 = 4913 and 65^2 = 4225. Since 4913 is greater than 4225, B is greater than A. Similar reasoning shows that C is greater than B. This confirms the ordering in statement III without relying on decimal approximations.

Why Other Options Are Wrong:

• Option a (Only I) suggests that A is largest, which contradicts the comparison of higher powers where C clearly dominates.
• Option c (Only II) proposes that B is largest and C is smallest, which is not supported by either estimation or exact power comparison.
• Option d (None of these) is incorrect because statement III is in fact correct.
• Option e (Both I and II) cannot be correct, since statements I and II conflict with each other.

Common Pitfalls:
A common mistake is to compare bases or exponents separately without considering the combined effect. For example, some might assume that the largest base must give the largest root, ignoring how a larger root index reduces the value. Another error is to attempt direct decimal calculations mentally, which can introduce serious rounding inaccuracies. Using comparison of higher powers with the same exponent is a more reliable strategy, especially for exam conditions where calculators are not allowed.

Final Answer:
The correct statement is Only III, that is, (12)^(1/3) > (17)^(1/4) > (65)^(1/6).