A maths question

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其實這個nested radical在MathWorld裡面已經可以找到答案。
從http://mathworld.wolfram.com/NestedRadical.html我們可以得知


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但是點解這個nested radical會 = x？
http://mathworld.wolfram.com/NestedRadical.html冇講，這表示我們需要自己證明。


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square it 10 times because there are 10 square roots
let the answer be a
(x^512)(x^256)(x^128)(x^64)(x^32)(x^16)(x^8)(x^4)(x^2)x=a^1024
x^(1-2^10/1-2)=a^10
x^1023=a^1024
a=x^(1023/1024)
so the answer is x^(1023/1024)

Let the answer be A

then

sqrt(xA) = A
xA = A^2
A = x

So the answer is x