To calculate this expression, we first need to simplify each term separately.

Log base 4 of 10 divided by log base 4 of 9:
log base 4 of 10 = log(10) / log(4)
log base 4 of 9 = log(9) / log(4)

Therefore, (log 10 / log 4) / (log 9 / log 4) = (log 10 / log 4) * (log 4 / log 9) = log(10) / log(9)

Log base 9 of 0.1:
log base 9 of 0.1 = log(0.1) / log(9)

Now substitute the values:

log(10) / log(9)log(0.1) / log(9)

Since log(10) = 1 and log(0.1) = -1, the expression simplifies to:

1 / log(9) + (-1) / log(9) = 0