SOLUTION: Without using a calculator or technology, determine which is bigger, {{{sqrt(2017)^(sqrt(2016))}}} or {{{sqrt(2016)^(sqrt(2017))}}}. Please prove your answer.

Algebra ->  Inequalities -> SOLUTION: Without using a calculator or technology, determine which is bigger, {{{sqrt(2017)^(sqrt(2016))}}} or {{{sqrt(2016)^(sqrt(2017))}}}. Please prove your answer.      Log On


   

Question 1045312: Without using a calculator or technology, determine which is bigger,
sqrt%282017%29%5E%28sqrt%282016%29%29 or sqrt%282016%29%5E%28sqrt%282017%29%29.
Please prove your answer.
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Without using a calculator or technology, determine which is bigger,
sqrt%282017%29%5E%28sqrt%282016%29%29 or sqrt%282016%29%5E%28sqrt%282017%29%29.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let x = sqrt%282017%29%5E%28sqrt%282016%29%29  and  y = sqrt%282016%29%5E%28sqrt%282017%29%29.


Then  

ln(x) = sqrt%282016%29%2Aln%28sqrt%282017%29%29 = %281%2F2%29%2Asqrt%282016%29%2Aln%282017%29    and  

ln(y) = sqrt%282017%29%2Aln%28sqrt%282016%29%29 = %281%2F2%29%2Asqrt%282017%29%2Aln%282016%29.


Then

ln%28x%29%2Fln%28y%29 = %28sqrt%282016%29%2Aln%282017%29%29%2F%28sqrt%282017%29%2Aln%282016%29%29 = %28%28sqrt%282016%29%2Fln%282016%29%29%29%2F%28%28sqrt%282017%29%2Fln%282017%29%29%29.

But  sqrt%28x%29%2Fln%28x%29 is monotonically increasing function, as everybody knows (or everybody should know who studies/studied Calculus).

Therefore,  ln%28x%29%2Fln%28y%29 = %28%28sqrt%282016%29%2Fln%282016%29%29%29%2F%28%28sqrt%282017%29%2Fln%282017%29%29%29 < 1.


It implies that  ln(x) < ln(y).


Hence,  x = sqrt%282017%29%5E%28sqrt%282016%29%29  <  y = sqrt%282016%29%5E%28sqrt%282017%29%29.


Answer.  sqrt%282017%29%5E%28sqrt%282016%29%29  <  sqrt%282016%29%5E%28sqrt%282017%29%29.

By the way, there is a classic question: which number is bigger (or larger, or greater):

e%5Epi or pi%5Ee.

The way to solve it is the same.


For more elementary inequalities see the lesson
    - What number is greater? Comparing magnitude of irrational numbers
in this site.


(one more time) By the way, you have free of charge online textbook in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.