You're dealing with a very simple unit conversion problem. Here's how to interpret such problems.
You were given a volume in "cm"^3 and asked to convert this volume to muL. To do this, you must be familiar with two things: "1 cm"^3 is equal to "1 mL" and that "1 L" is equal to 10^(3) "mL", and to 10^6 muL.
You can do this several ways, I"ll show you two ways.
1.08 * 10^(-23)"cm"^3 * ("1.0 mL")/("1.0 cm"^3) * (10^3muL)/("1.0 mL") = 1.08 * 10^(-20) muL
Notice that you go from "cm"^3 to "mL" using the first conversion factor and from "mL" to muL using the second one. You should really play around with going from one multiple to another, it's the only way to get good at doing this.
Here's another way you could have solved this
1.08 * 10^(-23)"cm"^3 * ("1.0 mL")/("1.0 cm"^3) * (10^(-3)"L")/("1.0 mL") * (10^(6)muL)/("1.0 L") = 1.08 * 10^(-20)muL
Same result, only this time you've converted to liters first, and then to microliters.