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24 Mar 2026, 05:00
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
51% (01:40) correct
49%
(01:25)
wrong
based on 98
sessions
History
Date
Time
Result
Not Attempted Yet
What is the smallest integer greater than \(\frac {5 + 3 }{√5 − √3}\)?
A) 13
B) 14
C) 15
D) 16
E) 17
A) 13
B) 14
C) 15
D) 16
E) 17
24 Mar 2026, 10:37
Kudos
Bookmarks
This is a rationalization problem. The core skill being tested: multiplying by the conjugate to clear the radical in the denominator.
1. The expression is 8 / (sqrt(5) - sqrt(3)). Rationalize by multiplying numerator and denominator by (sqrt(5) + sqrt(3)):
= 8(sqrt(5) + sqrt(3)) / ((sqrt(5))^2 - (sqrt(3))^2)
= 8(sqrt(5) + sqrt(3)) / (5 - 3)
= 8(sqrt(5) + sqrt(3)) / 2
= 4(sqrt(5) + sqrt(3))
= 4*sqrt(5) + 4*sqrt(3)
2. Now estimate. sqrt(5) is approximately 2.236 and sqrt(3) is approximately 1.732.
4 * 2.236 = 8.944
4 * 1.732 = 6.928
Total = 15.872
3. The smallest integer GREATER than 15.872 is 16.
Answer is D.
The trap here is answer C (15). People sometimes confuse "smallest integer greater than X" with "the integer closest to X." 15.872 rounds to 16, and the smallest integer strictly greater than 15.872 is also 16. But if someone computes 15.something and mistakenly stops at 15, they pick C. The question says "greater than," so 16 is correct.
I got burned by this exact wording trap early in my prep. The phrasing "smallest integer greater than" just means ceiling function. If the value were exactly 16.0, the answer would be 17 (since 16 would not be "greater than" itself). Good to keep that edge case in mind.
1. The expression is 8 / (sqrt(5) - sqrt(3)). Rationalize by multiplying numerator and denominator by (sqrt(5) + sqrt(3)):
= 8(sqrt(5) + sqrt(3)) / ((sqrt(5))^2 - (sqrt(3))^2)
= 8(sqrt(5) + sqrt(3)) / (5 - 3)
= 8(sqrt(5) + sqrt(3)) / 2
= 4(sqrt(5) + sqrt(3))
= 4*sqrt(5) + 4*sqrt(3)
2. Now estimate. sqrt(5) is approximately 2.236 and sqrt(3) is approximately 1.732.
4 * 2.236 = 8.944
4 * 1.732 = 6.928
Total = 15.872
3. The smallest integer GREATER than 15.872 is 16.
Answer is D.
The trap here is answer C (15). People sometimes confuse "smallest integer greater than X" with "the integer closest to X." 15.872 rounds to 16, and the smallest integer strictly greater than 15.872 is also 16. But if someone computes 15.something and mistakenly stops at 15, they pick C. The question says "greater than," so 16 is correct.
I got burned by this exact wording trap early in my prep. The phrasing "smallest integer greater than" just means ceiling function. If the value were exactly 16.0, the answer would be 17 (since 16 would not be "greater than" itself). Good to keep that edge case in mind.
