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# Help 2.0

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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 $$A + B + C + D + E + F$$?

May 15, 2019

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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

May 15, 2019