Listen to tonight's class, "Should You Make a Berakhah on Chewing Gum?" here.
Follow along with the sources here.
For further research:
1) Read some of the relevant sources in their entirety: a) Yabia Omer 7.33, b) Birkat HaShem 2, c) Yabia Omer 9.108, d) Birkat HaShem 5.
2) Read R. Ari Enkin's "Gum: Should a Blessing Be Recited?" here.