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2 square roots (Posted on 2023-03-12) Difficulty: 3 of 5
Evaluate, pen and paper only:

sqrt(sqrt(121)-sqrt(120))

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Solution Generalized Solution | Comment 3 of 14 |
121 is the square of an odd number, and 120 is the preceding number.  So I'll generalize this into sqrt[sqrt[(2k+1)^2]-sqrt[(2k+1)^2-1]].  Then the specific problem is evaluating this at k=5.

sqrt[sqrt[(2k+1)^2] - sqrt[(2k+1)^2-1]]
= sqrt[(2k+1) - sqrt[4k^2+4k]]
= sqrt[(k+1) - 2*sqrt[k*(k+1)] + k]
= sqrt[sqrt[k+1]^2 - 2*sqrt[k*(k+1)] + sqrt[k]^2]
= sqrt[(sqrt[k+1] - sqrt[k])^2]
= sqrt[k+1] - sqrt[k]

Then at k=5 we have sqrt[sqrt[121] - sqrt[120]] = sqrt[6] - sqrt[5].

  Posted by Brian Smith on 2023-03-12 17:31:27
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