Prove that 1/2+√3 + 2/√5-√3 + 1/2-√5=0

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Solution

(1/2+√3)+(2/√5-√3)+(1/2-√5)
=1(2-√3)/(2-√3)(2+√3)+2(√5+√3)/(√5+√3)(√5-√3)+1(2+√5)/(2+√5)(2-√5)
=(2-√3)/{2²-(√3)²}+2(√5+√3)/{(√5)²-(√3)²}+(2+√5)/{2²-(√5)²}
=(2-√3)/(4-3)+2(√5+√3)/(5-3)+(2+√5)/(4-5)
=2-√3+{2(√5+√3)/2}-(2+√5)
=2-√3+√5+√3-2-√5
=0 (Proved)
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