how to solve these type of questions ? i have tried logarithm and inequalities but could not pin point the exact and correct method.
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Such problems are based on relation of the surd with its conjugate
Where a,b, are integers. So it follows that
$(2+\sqrt3)^k+(2-\sqrt3)^k=2a $ is integer.
Now note that $(2-\sqrt3)^k<1$ implying $1-(2-\sqrt3)^k$ is fractional part of $(2+\sqrt3)^k$. So $1-f=(2-\sqrt3)^k$ and $x(1-f)=(2+\sqrt3)^k(2-\sqrt3)^k=1^k=1$