If √(12 − 2√35) + √(8 + 2√15) = √a + √b, where a and b are positive integers,…
2020
If √(12 − 2√35) + √(8 + 2√15) = √a + √b, where a and b are positive integers, then the value of ab is:
- A.
35
- B.
12
- C.
21
- D.
8
Attempted by 3 students.
Show answer & explanation
Correct answer: C
Idea: de-nest each radical using the identities x + y − 2√(xy) = (√x − √y)² and x + y + 2√(xy) = (√x + √y)².
First radical: √(12 − 2√35). Choose x and y with x + y = 12 and xy = 35; these are x = 7 and y = 5. So 12 − 2√35 = (√7 − √5)², and √(12 − 2√35) = √7 − √5 (positive root, since √7 > √5).
Second radical: √(8 + 2√15). Choose x and y with x + y = 8 and xy = 15; these are x = 5 and y = 3. So 8 + 2√15 = (√5 + √3)², and √(8 + 2√15) = √5 + √3.
Add them: (√7 − √5) + (√5 + √3) = √7 + √3. The √5 terms cancel.
Hence √a + √b = √7 + √3, so a = 7 and b = 3 (order does not matter). Therefore ab = 7 × 3 = 21.
Answer: 21.