Given ${\log }_{10}2=0.3010$

${\log }_{10}3=0.4771$

Now ${\log }_{10}{\left(2^{10}{3}^5\right)}^{20}=20{\log }_{10}\left(2^{10}{.3}^5\right)$

$=20({\log }_{10}{2}^{10}+{\log }_{10}{3}^5)$

$=20(10{\log }_{10}2+5{\log }_{10}3)$

$=20(10\times 0.3010+5\times 0.4771)$

$=20(3.010+2.3855)$

$=20\left(5.3955\right)$

$=107.91$

The number of digits in ${\left(2^{10}{3}^5\right)}^{20}=108$ digits