SOLUTION: I need help to solve: {{{ x^x = pi }}} to three decimal places.

There is no algebraic way to solve that.  You can solve it using
a TI graphical calculator, getting 1.854105968.  Or by trial and error.

Here is the trial and error method.


Try x=1

 = 

 = 

1 is too small

Try x=2

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 = 

2 is too large, so x is between 1 and 2.

Try x=1.5

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 = 

1.5 is too small, so x is between 1.5 and 2

Try x=1.7

 = 

 = 

1.7 is too small, so x is between 1.7 and 2 

Try x=1.9

 = 

 = 

1.9 is too large, so x is between 1.7 and 1.9

Try x=1.8

 = 

 = 

1.8 is too small, so x is between 1.8 and 1.9

Try x=1.85

 = 

 = 

1.85 is too small, so x is between 1.85 and 1.9

Try x=1.855

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 = 

1.855 is too large, so x is between 1.85 and 1.855

Try x=1.853

 = 

 = 

1.853 is too small, so x is between 1.853 and 1.855

Try x=1.854

 = 

 = 

1.854 is too small, so x is between 1.854 and 1.855

However we get closer to the right side when x = 1.854
than when x = 1.855

So the answer to three decimal places is x = 1.854.

Edwin