If the 4th term of ${\left\{\sqrt{\frac{1}{{\mathrm{x}}^{1+{\log }_{10}\mathrm{x}}}}+\sqrt[12]{\mathrm{x}}\right\}}^6$ is equal to $200, \mathrm{x}>1$ and the logarithm is common logarithm, then x is not divisible is

The given expansion is ${\left\{\sqrt{x^{\left(\frac{1}{1+{\log }_{10}x}\right)}}+\sqrt[12]{\mathrm{x}}\right\}}^6$

The general term in the expansion of ${\left(x+y\right)}^n$ is $T_{r+1}={}^{n}C_r{x}^{n-r}{y}^r$

Given fourth term of the expansion of ${\left\{\sqrt{x^{\left(\frac{1}{1+{\log }_{10}x}\right)}}+\sqrt[12]{\mathrm{x}}\right\}}^6$ is 200

Hence, ${}^{6}C_3{\left\{\sqrt{x^{\left(\frac{1}{1+{\log }_{10}x}\right)}}\right\}}^3{\mathrm{x}}^{\frac{1}{4}}=200$

Since the vlaue of ${}^{6}C_3$ is 20, cancel out 20 from both sides

${\left\{\sqrt{x^{\left(\frac{1}{1+{\log }_{10}x}\right)}}\right\}}^3{\mathrm{x}}^{\frac{1}{4}}=10$

it implies that $x=\boxed{10}$