Chets asked if 4+sqrt(5)*sqrt(3) = 1/(4-sqrt(5)*sqrt(3)) can be extrapolated into a broad generalization. No, it can't be!

Here's why.

Consider an expression

1/(a-\sqrt{bc}) [modeled on 1/(4-sqrt(5)*sqrt(3)) ]

Multiplying Numerator and Denominator with

(a+ \sqrt{bc}), we get

(a+ \sqrt{bc})/(a^2-bc)So, the reason why 1/(4-sqrt(5)*sqrt(3) got simplified into 4+sqrt(5)*sqrt(3) was because the Denominator of the above expression is equal to 1 in this particular case (a=4, b=5, c=3. So,

a^2-bc = 4^2-5*3 = 1)

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