if a=2+√3 find a-1/a
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a=2+√3
1/a= 1/(2+√3)
1/a= (2-√3)/(2+√3)(2-√3)
1/a= (2-√3)/(4-3)
1/a= (2-√3)
so, (a-1/a)= (2+√3)-(2-√3) = 2√3
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