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Given that 4 / (2 + sqrt(3) + sqrt(7)) = sqrt(a) + sqrt(b) - sqrt(c), which of the following sets of values (a, b, c) can be true?

1. a = 1, b = 4/3, c = 7/3
2. a = 1, b = 2/3, c = 7/9
3. a = 2/3, b = 1, c = 7/3
4. a = 7/9, b = 4/3, c = 1

Correct answer: a = 1, b = 4/3, c = 7/3

Solution

Rationalizing gives (2 + sqrt(3) - sqrt(7)) / sqrt(3) = 2/sqrt(3) + 1 - sqrt(7/3) = sqrt(4/3) + sqrt(1) - sqrt(7/3). Since addition is commutative, setting a=1, b=4/3, c=7/3 gives sqrt(1) + sqrt(4/3) - sqrt(7/3), which equals the same expression.

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