SOLUTION: Evaluate https://imgur.com/a/VNO3CQS Equation was to complicated to write.

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Question 1151253: Evaluate https://imgur.com/a/VNO3CQS
Equation was to complicated to write.
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solution by tutor @MathLover1 is fine until the final answer; apparently the final "simplified" expression was evaluated incorrectly.

When the expression is simplified in the manner shown in his solution, you end up with an expression that is only slightly less complicated than the original expression; and you still need to use a calculator to evaluate the final expression.

So you might as well just use your calculator to evaluate the original expression:

sqrt%285%2B%28sqrt%28%285%5E2%29%2B%28sqrt%28%285%5E4%29%2Bsqrt%285%5E8%29%29%29%29%29%29 = 3.573355548 to 9 decimal places.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Actually,  the problem asks about  OTHER  expression,  which is  infinite sequence  of square roots (!)

            It is  different  from that considered by two other tutors,  so their solutions and answers are not relevant.

            For the right solution,  see below. 


Let  

    x = sqrt%285%2Bsqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bellipsis%29%29%29%29      (1) 

be the value to determine for.


Square both sides.  You will get

    x%5E2 = 5 + sqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bsqrt%285%5E16%2Bellipsis%29%29%29%29,

or

    x%5E2+-+5 =  sqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bsqrt%285%5E16%2Bellipsis%29%29%29%29   (2)


The right side of (2) can be written in this form


    sqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bsqrt%285%5E16%2Bellipsis%29%29%29%29 = sqrt%285%29*sqrt%285%2Bsqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bellipsis%29%29%29%29,


which is the same as


    sqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bsqrt%285%5E16%2Bellipsis%29%29%29%29 = sqrt%285%29%2Ax.


Therefore, equation (2) takes the form


    x%5E2+-+5 = sqrt%285%29%2Ax,   or


    x%5E2+-+sqrt%285%29%2Ax+-+5 = 0.    (3)


Now apply the quadratic formula to get the roots of the equation (3)


    x%5B1%2C2%5D = %28%28sqrt%285%29%29+%2B-+sqrt%28+%28sqrt%285%29%29%5E2+-+4%2A%28-5%29%29%29%2F2 = %28sqrt%285%29+%2B-+sqrt%285+%2B+20%29%29%2F2 = %28sqrt%285%29+%2B-+sqrt%2825%29%29%2F2 = %28sqrt%285%29+%2B-+5%29%2F2.


We ignore the negative root and accept only positive value


    x = %28sqrt%285%29+%2B+5%29%2F2.


It is the ANSWER  to the problem's question.


The approximate value is  x = %28sqrt%285%29+%2B+5%29%2F2 = 3.618034... 


    For checking purposes, the value of the truncated expression   sqrt%285%2Bsqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%29%29%29%29  is  3.573356...


ANSWER.  The solution is  x = sqrt%285%2Bsqrt%285%5E2%2Bsqrt%285%5E4%2Bsqrt%285%5E8%2Bellipsis%29%29%29%29 = %28sqrt%285%29+%2B+5%29%2F2 = 3.618034...

Solved.

------------

See the lesson
    - Evaluating expressions that contain infinitely many square roots
in this site and find there the solutions to many othe similar problems.

All these problems,  including the current one,  are of the  Math circle  level.