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

Question 1045312: Without using a calculator or technology, determine which is bigger,
or .
Please prove your answer.
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
.
Without using a calculator or technology, determine which is bigger,
or .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let x =   and  y = .


Then  

ln(x) =  =     and  

ln(y) =  = .


Then

 =  = .

But   is monotonically increasing function, as everybody knows (or everybody should know who studies/studied Calculus).

Therefore,   =  < 1.


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


Hence,  x =   <  y = .


Answer.    <  .

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

or .

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.

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