Large indeed!

Algebra Level 2
  • 2 2 2 2^{2^{2}} is 2 digits long.
  • 3 3 3 3^{3^{3}} is 13 digits long.
  • 4 4 4 4^{4^{4}} is 155 digits long.
  • How long is 9 9 9 9^{9^{9}} ?


The answer is 369693100.

Gabriel Chacón by Gabriel Chacón , 55, Spain

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1 solution

Gabriel Chacón
Jan 2, 2019

To count the digits, we express the number as a power of 10. The exponent rounded up to the nearest integer yields the number of digits: log ⁡ ( 9 9 9 ) = 9 9 log ⁡ 9 ≈ 369693099.6 \log{(9^{9^9})}={9^9\log9} \approx 369693099.6 . The number has 369 693 100 \boxed{369\,693\,100} digits!

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