How do I "determine the mass percent of H2O in the hydrate Cr(NO3)3 * 9H2O to 3 significant figures"?

Answer 1

It depends if you are determining from mass measurements, or a formula calculation.

I assume you took an experment, measured some of the hydrate, heated it, then measured the unhydrated salt.

In that case,

Percenthydrate= 100* (masshydrate-massunhydrated)/masshydrated

Now, to do this to three sig figures might be a problem, as the number of sig digits might be determined by the subtracation, rather than your wishes.
For instance, say
mass hydrated= 10.34 grams
mass unhydrage= 9.51 grams

subtracting, one gets .83 grams, and you cannot get three sig figures from this.

Answer 2

On the other hand, if you are calculating percent water from the formula, then,
[9*molar mass H2O/molar mass Cr(NO3)3*9H2O]*100 = %H2O