Given x = square root (3-square root 5) + square root(3+square root 5) prove that x^2 = 10

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We have x = sqrt(3 - sqrt 5) + sqrt(3 + sqrt 5).

x^2 = [sqrt(3 - sqrt 5) + sqrt(3 + sqrt 5)]^2

=> (3 - sqrt 5) + (3 + sqrt 5) + 2*sqrt(3 - sqrt 5)*sqrt(3 + sqrt 5)

=> 6 + 2*sqrt(9 - 5)

=> 6 + 2*2

=> 10

This proves that x^2 = 10

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