Don't do it the long way

Algebra Level 3

What is the correct order of these expressions, from smallest to largest?

  • 3 3 4 \large 3^{3^4}

  • 3 4 3 \large 3^{4^3}

  • 3 4 4 \large 3^{4^4}

  • 4 3 3 \large 4^{3^3}

  • 4 3 4 \large 4^{3^4}

Give your answer as the number of the smallest, followed by second smallest, and so on until the largest.

e.g. If you think they are already in order, then your answer should be 12345.


The answer is 42153.

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1 solution

The quickest way to solve this problem is to consider the exponents. remember that 3^3^4 means 3^(3^4) or 3^81. We can do this for each giving us 3^81, 3^64, 3^256, 4^27 and 4^81.

we now have two obvious sets of inequalities: 3^64<3^81<3^256 and 4^27<4^81.

The next step is less obvious but if we write 4 as 2^2 we now have \2 ( 2 27 ) < \2 ( 2 81 ) \2^{(2* 27)}< \2^{(2* 81)} , we can now deduce that 4^3^3 is the smallest as 2^54 is obviously less that 3^64. So we now have 4^3^3 < 3^4^3 < 3^3^4 < 3^4^4.

All that is left to do is find where 4^3^4 fits in. We can see that it will be greater than 3^3^4 as a greater (positive) base to the same power will always be greater. We can also see (from 4^3^4=2^162) that it is less than 3^4^4=3^256 as a lower positive base to a lower power will always be lower.

This leaves the final inequality as 4^3^3<3^4^3<3^3^4<4^3^4<3^4^4, subbing the numbers back in gives 42153

Great solution William! I've gone ahead and cleaned up the LaTeX on this problem as well as on your problem Balancing Act . If you'd like to see what I did, click "edit problem" in the hamburger menu in the upper right. :)

Andrew Ellinor - 5 years, 1 month ago

Loved solving this question

Abhiram Rao - 5 years, 1 month ago

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