You can split the fraction as it follows:

#\frac{1.1 * 10^22}{6.022 * 10^23} = \frac{1.1}{6.022}\ \frac{10^22}{10^23}#

and deal with the two parts separately.

#\frac{1.1}{6.022}# is a common numeric division, and the result is #0.1827#

As for #\frac{10^22}{10^23}#, you can see it in two ways: either you expand the powers, having a fraction of the form

#\frac{10 * 10 * ... * 10}{10 * 10 * 10 * ... * 10}#, with 22 "tens" at the numerator and 23 at the denominator. So, you can simplify all the "tens", except one at the denominator, obtaining #10^22/10^23=1/10#.

In a more formal fashion, you have that #10^22/10^23=10^{-1}#, which is again #1/10#.

So, the final answer is #\frac{1.1 * 10^22}{6.022 * 10^23} = 0.1827 * 10^{-1}#.