Question 1016749
<pre>
Let the expression = x

{{{ sqrt(4+sqrt(4- red(sqrt( 4+ sqrt(4-""*""*""*"")))))}}}{{{""=""}}}{{{x}}}

Notice that the red expression is the same as the original 
expression, so

{{{ sqrt(4+sqrt(4- x))))}}}{{{""=""}}}{{{x}}}

Square and isolate radicals until you end up with this 4th degree equation.

{{{x^4-8x^2+x+12}}}{{{0}}}{{{""=""}}}{{{0}}}

which factors by trial and error as

{{{(x^2-x-3)(x^2+x-4)}}}{{{""=""}}}{{{0}}}

Setting each one equal to 0 and solving by the quadratic formula
gives 

{{{x=(1 +- sqrt(13))/2}}} and {{{x=(1 +- sqrt(17))/2}}}

The answer must be positive, so it is one of these:

{{{x=(1 + sqrt(13))/2}}} or {{{x=(1 + sqrt(17))/2}}}

The first one satisfies

{{{ sqrt(4+sqrt(4- x))))}}}{{{""=""}}}{{{x}}}

but the second one does not.

Answer: {{{ sqrt(4+sqrt(4- sqrt( 4+ sqrt(4-""*""*""*""))))}}}{{{""=""}}}{{{(1 + sqrt(13))/2}}}

Edwin</pre>