Largest among 100^1/100, 100^1/101, 10^1/51, 10^1/52: Detailed Comparison Explained

Introduction

Questions asking which is the largest among expressions like
100^(1/100), 100^(1/101), 10^(1/51), and 10^(1/52)
are common in competitive exams and test conceptual understanding of exponents rather than direct computation.

By using monotonicity properties of functions of the form a^(1/n) and comparing bases and exponents intelligently,
this problem can be solved quickly without calculators.

Key Concept: Behavior of a^1/n

For a fixed base a > 1, the function
f(n) = a^(1/n) decreases as n increases, because
ln(f(n)) = (ln a)/n becomes smaller when n grows.

Therefore, with the same base, the smaller denominator in the exponent gives the larger value of a^(1/n).

Given Options

The four options in the question are:

• (a) 100^1/100
• (b) 100^1/101
• (c) 10^1/51
• (d) 10^1/52

Comparing Options with Base 100

Compare options (a) and (b): both have base 100, but the exponents are 1/100 and 1/101 respectively.
Since 100 > 1 and 1/100 > 1/101, and 100^1/n decreases as n increases, we have:

100^1/100 > 100^1/101.

Hence, among (a) and (b), option (a) is larger.

Comparing Options with Base 10

Compare options (c) and (d): both have base 10, with exponents 1/51 and 1/52 respectively.
Since 10 > 1 and 1/51 > 1/52, and 10^1/n also decreases as n increases, we obtain:

10^1/51 > 10^1/52.

Therefore, among (c) and (d), option (c) is larger.

Main Comparison: 100^1/100 vs 10^1/51

From the previous comparisons, the real contenders for the overall largest value are (a) 100^1/100
and (c) 10^1/51.

Rewrite 100 in terms of base 10: 100 = 10². Therefore,

100^1/100 = (10²)^1/100 = 10^2/100 = 10^1/50.

Now compare 10^1/50 and 10^1/51: both share the same base 10,
and 1/50 > 1/51, so

10^1/50 > 10^1/51.

Hence, 100^1/100 = 10^1/50 is greater than 10^1/51 and therefore
greater than options (c) and (d), and it was already greater than option (b).

Final Result

Combining all these comparisons:

100^1/100 > 100^1/101,
10^1/51 > 10^1/52, and
10^1/50 > 10^1/51.

Therefore, the largest among all four options
100^1/100, 100^1/101, 10^1/51, and 10^1/52
is 100^1/100, i.e., option (a).

Option-wise Summary

• Option (a) 100^1/100:
  Equal to 10^1/50; largest value due to the highest effective exponent on base 10.
• Option (b) 100^1/101:
  Smaller than (a) because the denominator 101 is larger, making the exponent smaller.
• Option (c) 10^1/51:
  Larger than (d) but smaller than (a) since 1/51 is less than 1/50.
• Option (d) 10^1/52:
  Smallest of all because it has the largest denominator in the exponent for base 10.