Alternative Dosing of Exemestane Before Surgery in Treating Postmenopausal Patients With Stage 0-II Estrogen Positive Breast Cancer

Inclusion Criteria:

* Postmenopausal women (postmenopausal: age \>= 60 years, or amenorrhea \>= 12 months, or bilateral oophorectomy, or - in women with hysterectomy only - follicle stimulating hormone \[FSH\] in the menopausal levels as per local institutional guidelines if \< 60 years old) with histologically-confirmed estrogen receptor (ER)-positive (\>= 10%) primary breast cancer stage cT0-2, cN0-1, Mx; women with larger tumors who refuse chemotherapy (chemo) and/or endocrine neoadjuvant therapy can be eligible
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%)
* Leukocytes \>= 3,000/microliter
* Absolute neutrophil count \>= 1,500/microliter
* Platelets \>= 100,000/microliter
* Total bilirubin =\< 2 x institutional upper limit of normal (ULN)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 1.5 x institutional ULN
* Serum creatinine =\< 1.5 times institutional ULN
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Body mass index (BMI) \< 18.5 Kg/m\^2
* Previous treatment for breast cancer including chemotherapy, endocrine therapy and radiotherapy; women with prior ductal breast carcinoma in situ (DCIS) who were treated with surgery only and whose treatment ended \>= 2 years prior to enrollment are eligible for the trial
* Women who are planned to receive neoadjuvant therapy
* Participants may not be receiving investigational agents
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to exemestane
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Other co-existing invasive malignancies (with the exclusion of basal cell carcinoma or skin squamous cell carcinoma) diagnosed during the last 2 years before randomization
* History of severe osteoporosis (T score =\< -4 either spine or hip), or presence of vertebral fracture
* Use of systemic hormone replacement therapy (HRT) in the last 30 days prior to the randomization; the use of non-systemic estrogen (such as vaginal estrogen use) is allowed
* Use of any chemopreventive agents (selective estrogen receptor modulators \[SERM\]) in the last 3 months
* Concomitant use of CYP3A4 inducer medication (rifampicin, phenytoin, carbamazepine, phenobarbital, and St. John's wort)