(R)-3-Bromo-2,4-dimethylpenta... reacts with sodium hydroxide to yield a mixture of R and S alcohols?

Well you didn't get to write the full name of the compound, but I'm guessing that this would be an SN1 nucleophilic addition reaction (try looking up sn1 in Wikipedia). SN2 would only give you R or S enantiomers.

Um, if you don't know what an R/S enantiomer is... you should make your product with a molecular model kit. There will be a carbon in there with 4 different groups attached to it (eg an OH, an O-CH3, a CH2CH2CH2CH3 and an H). Number these in order of the atom with the biggest atomic number attached (keep going along the chain if two atoms are the same). For this eg you would have:
1. O-CH3
2. OH
3. CH2CH2CH2CH3
4. H

Now hold your model with the number 4 group furthest away from you. If the numbers 1, 2, 3 are clockwise, then it's an R conformation; anti-clockwise and it's an S conformation.


So if your starting compound has a double bond, it will add an Na first then an OH. Might need to check whether the reaction is Markovnikov or anti-Markovnikov to know which side the Na and OH add to (if you've never heard of that word then ignore it?).