I would avoid floating point exponents like .97 or .8 and do as much of the calculation with exact numbers as possible. If Mathematica is not wrong, Q seems to be .7934:
$MaxExtraPrecision = 10000;
{ Denominator[BigFrac]^(7934/10000) (Pi + BigFrac) > 1,
Denominator[BigFrac]^(7945/10000) ( Pi + BigFrac) > 1}
