Prove that the function $e^{-\sqrt{|\ln\text{frac}(x)|}}-(\text{frac}(x))^{\sqrt{\frac{1}{|\ln\text{frac}(x)|}}}$ is both even and odd function.

Question

Prove that the function $e^{-\sqrt{|\ln\text{frac}(x)|}}-(\text{frac}(x))^{\sqrt{\frac{1}{|\ln\text{frac}(x)|}}}$ is both even and odd function.Here $\text{frac}(x)$ is a fractional part function.

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If i need to prove a function both even and odd function,means i have to prove it zero function,because only zero function is both even and odd function.

I found the domain of the function it is $R-Z$,all real numbers excluding integers.
But i cannot prove $e^{-\sqrt{|\ln\text{frac}(x)|}}=(\text{frac}(x))^{\sqrt{\frac{1}{|\ln\text{frac}(x)|}}}$

Please help me.Thanks.