"the compound k2o2 also exists. a chemist can determine the mass of k in a sample of known mass that consists of either pure k2o or pure k2o2. from this information, can the chemist answer the question of which compound is in the sample

Yes, the chemist can determine which compound is in the sample.

Explanation:

In 1 mole of K₂O, the mass of K is 2 × 39.1 g = 78.2 g and the mass of K₂O is 94.2 g. The mass ratio of K to K₂O is 78.2 g / 94.2 g = 0.830.

In 1 mole of K₂O₂, the mass of K is 2 × 39.1 g = 78.2 g and the mass of K₂O₂ is 110.2 g. The mass ratio of K to K₂O₂ is 78.2 g / 110.2 g = 0.710.

If the chemist knows the mass of K and the mass of the sample, he or she must calculate the mass ratio of K to the sample.

it's all four options.