[Math] Number of zeros between decimal and first significant digit

Question

How can we determine number of zeros between decimal and first significant digit in $\big(\frac{1}{6}\big)^{200}$?

I can't develop any fruitful approach for this question. Could some provide some hint?

Answer 1

A number $x$ satisfies $$10^{-n} \le x \le 10^{-n+1}$$ if and only if $$-n \le \log_{10} x \le -n+1$$ Now, $$\log_{10} 6^{-200} = -200 \log_{10} 6 = -155.6... < -155$$ so that your number has $155$ zeroes after the point.