April 16th, 2014, 04:17 AM  #1 
Senior Member Joined: Nov 2013 Posts: 434 Thanks: 8  solve
solve x^x^x^16 =16 
April 16th, 2014, 09:13 AM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,412 Thanks: 1024  
April 16th, 2014, 09:21 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,616 Thanks: 2072 
x^x^x^16 means x^(x^(x^16)), not (x^x^x)^16. Carry on...

April 16th, 2014, 07:57 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,412 Thanks: 1024  I had written short numeric program; x^x^x^16 = 16 ; x = ~1.14208 Just went back and changed to what you show: x^(x^(x^16)) = 16 ; x = ~1.18921 I'm surprised that the change from x^x^x^16 to x^(x^(x^16)) is not automatic (not in UBasic anyway). So shouldn't the OP have included the brackets then? Thanks for stepping in Skip; can you clarify (I'm also a learner!). 
April 16th, 2014, 10:46 PM  #5 
Newbie Joined: Apr 2014 From: india Posts: 3 Thanks: 0 
$\displaystyle x=1.1892071150027$
Last edited by mhacker064; April 16th, 2014 at 10:48 PM. 
April 17th, 2014, 12:01 AM  #6 
Senior Member Joined: Apr 2014 From: UK Posts: 914 Thanks: 331 
I took the original to mean ((x^x)^x)^16 = 16 giving x =~1.14208 I believe this is the correct answer because you work from left to right (and that's what the web tells me too ) 
April 17th, 2014, 02:19 AM  #7 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
I agree with Skipjack: the expression should be interpreted as $\displaystyle x^{x^{x^{16}}}$. If we wanted it to mean the other one, we could have written $\displaystyle x^{16x^2}$ instead.

April 17th, 2014, 03:34 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,616 Thanks: 2072 
The usual interpretation regarding order of operations is what I gave, as the alternative interpretation allows easy simplification. For this problem, it turns out that one can add extra occurrences of "x^" without affecting the final answer.
Last edited by skipjack; April 17th, 2014 at 03:38 AM. 
April 17th, 2014, 06:02 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,412 Thanks: 1024 
Interesting stuff... Using k=x^x^x then k = 16^(1/16) = 1.1892071... Now we have x^x^x = 1.1892071, so x = 1.1420819... In UBasic: x = 1.1420819 PRINT x^x^x^16 Output: ~16 Looks like that in programming languages, the "brackets" need to be included. 
April 17th, 2014, 06:25 AM  #10 
Senior Member Joined: Apr 2014 From: UK Posts: 914 Thanks: 331 
This is why I like brackets.... Wolfram agrees with it being x^(x^(x^16)) This page has an example with the right answer, but the wrong explanation! Order of Operations  GMAT Math Study Guide problem 2 at the bottom... 