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.