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Without computing the decimal values of sqrt(3) and sqrt(14), determine which of the two numbers 5 + sqrt(3) and 3 + sqrt(14) is greater.

1. 5 + sqrt(3)
2. 3 + sqrt(14)
3. They are equal
4. Cannot be determined

Correct answer: 3 + sqrt(14)

Solution

Subtract 3 from both: compare 2 + sqrt(3) with sqrt(14). Both positive, so square: (2 + sqrt(3))² = 4 + 4sqrt(3) + 3 = 7 + 4sqrt(3) versus 14. Compare 4sqrt(3) with 7 by squaring: 16*3 = 48 versus 49. Since 48 < 49, 4sqrt(3) < 7, so 7 + 4sqrt(3) < 14, meaning 2 + sqrt(3) < sqrt(14), thus 5 + sqrt(3) < 3 + sqrt(14). Therefore 3 + sqrt(14) is the greater number.

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