Create The Largest Numbers With 3 Digits!

Algebra Level 1

Which of the following numbers is the largest?

9 9 9 9^{9^9} ( 9 9 ) 9 (9^9)^9 999 999 9 9 9 99^9

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Jeff Carter by Jeff Carter , 59, USA

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1 solution

Sam Bealing
Apr 7, 2016

Consider the expressions log ⁡ 9 \log_{9} :

  • log ⁡ 9 ( 999 ) < log ⁡ 9 ( 6561 ) = 4 \log_{9}(999)<\log_{9}(6561)=4
  • log ⁡ 9 ( 9 9 9 ) = 9 log ⁡ 9 ( 99 ) < 9 log ⁡ 9 ( 729 ) = 27 \log_{9}(99^9)=9 \log_{9}(99)<9 \log_{9}(729)=27
  • log ⁡ 9 ( 9 9 9 ) = 9 9 \log_{9}(9^{9^9})=9^9
  • log ⁡ 9 ( ( 9 9 ) 9 ) = 9 log ⁡ 9 ( 9 9 ) = 81 = 9 2 \log_{9}((9^9)^9)=9\log_{9}(9^9)=81=9^2

But clearly 9 2 < 9 9 9^2<9^9 so 9 9 9 9^{9^9} is the biggest number.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

When attempting to compare large numbers, it is sometimes easier to look at their approximate logarithmic value (for a suitable base).

For example, comparing the number of digits in the number is simply working in logarithmic base 10.

Well that's complicated. Just look at the numbers. 9^9^9 seems to be the highest.. so click it. :'D

Dilara Merve - 5 years, 1 month ago

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But that wouldn't really be a solution!

Sam Bealing - 5 years, 1 month ago

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