Ginger root

I have two large ( each piece is larger than my hand with fingers spread) of ginger root.  I am trying to figure out how to keep them. The only thing I am coming up with is dehydration.  What are other options?

How long do you want to keep them? I often keep the fresh root in the fridge, sealed in a small container for quite a long time - 2 - 3 months, in fact, without molding. If you want to keep it longer, without dehydrating, freezing is a great way to go. You can also ferment it - and I keep a lemon-ginger-honey ferment on my counter all the time. It's great in tea, drizzled over ice cream or bread pudding, or as an ingredient in Asian cooking or mixed with a bit of red pepper flakes, as a dip or drizzle for fish tacos, pork tenderloin, or chicken. But, the main reason I do that, is it's the most effective cough syrup I've tried.

My favourite way of preserving ginger is to julienne finely and dehydrate.

It can then be used to make a hot lemon and ginger tea, added to marinades, chopped even more finely and added to chutneys.

if you plan to cook or make tea with it, I agree with Carla, stick it in the fridge drawer, it should keep a while.
In my house, my husband is the main consumer. We buy kilos at a time to make fiery candied ginger, which is his basic cure-all for colds and such that go around. While he does that, I make pickled ginger (gari and benishoga), which we eat alongside our meals. All the skins and off-cuts that don't make it into these things get juiced, and I freeze the juice in ice cube trays for cooking and tea (I recommend Japanese pork with cabbage and ginger, butashogayaki), but if my kid wants it sometimes I'll boil these skins and trimmings to make ginger syrup, which she'll mix with soda water to make homemade ginger ale.
Another good thing to do is make a ginger bug (fermented drink).

another idea, if you plan to use it for cooking or tea, to save Future You some headaches: make paste and freeze it. https://ministryofcurry.com/ginger-paste/ .
It's a relatively common ingredient in Indian cooking; I've made the version that is ginger and garlic together, it's very handy if you make this kind of food often.

(I always have fresh ginger on hand, so don't make just the ginger paste, but I have also heard of freezing the entire root. Not sure how that affects how you can use it when you defrost)