Approximately 4.8989795
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1/(5+sqrt(6)) is equal to about .134237
You rationalize denominators by using a form of one. Thus. sqrt(X + 6)/sqrt(X) * sqrt(X)/sqrt(X) = sqrt(X2 + 6X)/X ------------------------
11*sqrt(8) + 6*sqrt(12) - 5*sqrt(2) = 11*sqrt(4*2) + 6*sqrt(4*3) - 5*sqrt(2) = 11*sqrt(4)*sqrt(2) + 6*sqrt(4)*sqrt(3) - 5*sqrt(2) = 11*2*sqrt(2) + 6*2*sqrt(3) - 5*sqrt(2) = 22*sqrt(2) - 5*sqrt(2) + 12*sqrt(3) = 17*sqrt(2) + 12*sqrt(3) and that cannot be simplified further.
sqrt(24) = sqrt(4*6) = sqrt(4)*sqrt(6) = 2*sqrt(6)
sqrt(54) = sqrt(9*6) = sqrt(9)*sqrt(6) = 3*sqrt(6)