How do I simplify radical equations with nested radicals?

[Agent_J]

Simplify
sqrt(11+sqrt72)) + sqrt(11-sqrt(72))

better picture here
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I don't know where to begin for this one, but apparently my calculator says the answer is 6

 

[matt grime]

let x be the surd, square x, what do you get?

 

[Agent_J]

let x be the "surd"?

 

[deltabourne]

ok, set sqrt(11+sqrt72)) + sqrt(11-sqrt(72)) = x
(by the way, sqrt(72) = 6*sqrt(2), which i'll just call 6r2 for simplicity)
square both sides, we get
(sqrt(11+sqrt72)) + sqrt(11-sqrt(72)))^2 = x^2
simplify and you get
11+6r2+2*sqrt((11+6r2)(11-6r2))+11-6r2=x^2
22+2sqrt(121+11*6r2-11*6r2-36*2)=x^2
22+2*sqrt(121-72)=x^2
22+2*sqrt(49)=x^2
22+2*7=x^2
36=x^2
x=6 (well, plus or minus, but we know it must be positive since the addition of two non-complex roots must be >= 0)

 

[matt grime]

Yes, surd: an expression involving radicals. I just didn't want to have to type it out. Just square the expression, simplify and take the square root, et voila, we have 6, with the appropriate choice of sign.