Rationalize the denominator of \(\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}\), and write your answer in the form \( \frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F},\) where everything is in simplest radical form and the fraction is in lowest terms. What is \(A + B + C + D + E + F\)?
+128475 1 (sqrt(2) + sqrt(3) - sqrt(7)
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(sqrt (2) + sqrt(3) + sqrt(7) ) (sqrt(2) + sqrt(3) - sqrt(7))
(sqrt(2) + sqrt(3) -sqrt(7)
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2 + sqrt(6) - sqrt(14) + sqrt(6) + 3 - sqrt(21) + sqrt(14) + sqrt(21) - 7
sqrt(2) + sqrt(3) - sqrt(7) (2sqrt(6) + 2)
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(2sqrt(6) - 2) (2sqrt(6) + 2)
2sqrt(12) + 2sqrt(2) + 2sqrt(18) + 2sqrt(3) - 2sqrt(42) - 2sqrt(7)
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24 - 4
2sqrt(12) +2sqrt(2) + 2sqrt(18) + 2sqrt(3) - 2sqrt(42) - 2sqrt(7)
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20
sqrt(12) + sqrt(2) + sqrt(18) + sqrt(3) - sqrt(42) - sqrt(7)
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10
2sqrt(3) + sqrt(2) + 3sqrt(2) + sqrt(3) - sqrt(42) - sqrt (7)
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10
4sqrt(2) + 3sqrt(3) - 1sqrt(7) - 1sqrt(42)
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10
A + B + C + D + E + F =
4 + 3 - 1 - 1 + 42 + 10 =
57
