SOLUTION: {{{ sqrt(4+sqrt(4- sqrt( 4+ sqrt(4-""*""*""*""))))}}}

Algebra ->  Square-cubic-other-roots -> SOLUTION: {{{ sqrt(4+sqrt(4- sqrt( 4+ sqrt(4-""*""*""*""))))}}}      Log On


   



Question 1016749: +sqrt%284%2Bsqrt%284-+sqrt%28+4%2B+sqrt%284-%22%22%2A%22%22%2A%22%22%2A%22%22%29%29%29%29
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Let the expression = x

%22%22=%22%22x

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

+sqrt%284%2Bsqrt%284-+x%29%29%29%29%22%22=%22%22x

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

x%5E4-8x%5E2%2Bx%2B120%22%22=%22%220

which factors by trial and error as

%28x%5E2-x-3%29%28x%5E2%2Bx-4%29%22%22=%22%220

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

x=%281+%2B-+sqrt%2813%29%29%2F2 and x=%281+%2B-+sqrt%2817%29%29%2F2

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

x=%281+%2B+sqrt%2813%29%29%2F2 or x=%281+%2B+sqrt%2817%29%29%2F2

The first one satisfies

+sqrt%284%2Bsqrt%284-+x%29%29%29%29%22%22=%22%22x

but the second one does not.

Answer: +sqrt%284%2Bsqrt%284-+sqrt%28+4%2B+sqrt%284-%22%22%2A%22%22%2A%22%22%2A%22%22%29%29%29%29%22%22=%22%22%281+%2B+sqrt%2813%29%29%2F2

Edwin