How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^5\cdot5^8}$?
To answer this question, I need to convert the fraction $\frac{1}{2^5\cdot5^8}$ into a decimal. I know that dividing by a power of 10 shifts the decimal point to the left by the same number of places as the exponent. So, if I multiply the numerator and denominator of the fraction by $2^3$, I get $\frac{2^3}{2^8\cdot5^8}=\frac{8}{10^8}$. This means that the decimal representation of the fraction is $0.00000008$. Now I can count how many 0's are between the decimal point and the first non-zero digit. I see that there are 7 such 0's. 