Question 118728
I need help to solve: {{{ x^x = pi }}} to three decimal places.
<pre><font face = "book antiqua" size = 5><b>
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

{{{1^1}}} = {{{3.141592654}}}

{{{1}}} = {{{3.141592654}}}

1 is too small

Try x=2

{{{2^2}}} = {{{3.141592654}}}

{{{4}}} = {{{3.141592654}}}

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

Try x=1.5

{{{1.5^1.5}}} = {{{3.141592654}}}

{{{1.837117307}}} = {{{3.141592654}}}

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

Try x=1.7

{{{1.7^1.7}}} = {{{3.141592654}}}

{{{2.464694899}}} = {{{3.141592654}}}

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

Try x=1.9

{{{1.9^1.9}}} = {{{3.141592654}}}

{{{3.385570344}}} = {{{3.141592654}}}

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

Try x=1.8

{{{1.8^1.8}}} = {{{3.141592654}}}

{{{2.880650097}}} = {{{3.141592654}}}

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

Try x=1.85

{{{1.85^1.85}}} = {{{3.141592654}}}

{{{3.120812649}}} = {{{3.141592654}}}

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

Try x=1.855

{{{1.855^1.855}}} = {{{3.141592654}}}

{{{3.146139392}}} = {{{3.141592654}}}

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

Try x=1.853

{{{1.853^1.853}}} = {{{3.141592654}}}

{{{3.135979045}}} = {{{3.141592654}}}

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

Try x=1.854

{{{1.854^1.854}}} = {{{3.141592654}}}

{{{3.141054263}}} = {{{3.141592654}}}

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</pre>