The fraction $\dfrac{3^{2016}}{4^{2017}5^{2018}}$ can be converted into a terminating decimal number. Excluding the optional $0$s at the right end, how many digits are there to the right of the decimal point in the decimal number corresponding to that fraction?

I've gotten the answer 2^2016 + 2, but that's unrealistic because that's not solvable. I noticed $\dfrac{3^{n}}{4^{n}5^{n}}$ will always result in $2^{n}$. The extra 4^2(5) is 100 which will add 2 decimal places. Unless that's wrong. What's the correct answer? And if you can, explain what I did wrong? (You don't have to)

Thanks so much for helping me!

Guest Oct 7, 2020

#2**0 **

(3^2016) / (4^2017 * 5^2018) =9999499344264..........43212890625 =2371 digits excluding extraneous zeros to the right of the decimal point.

Guest Oct 7, 2020

#3**0 **

Just to expand on the above and your observation that: 3^n /[4^n*5^n] =2^n. The answer is NOT 2^n but 2*n. So: We can write your expression as follows: 3^2016 / [4 * 4^2016 * 5^2 * 5^2016] =2 * 2016 + 2=4034 digits after the decimal point, which is correct indeed! There are 1663 zeros to the right of the decimal point + 2371 other digits = 4034 digits after the decimal point. I had this confirmed by writing a short computer code in C++, which printed ALL 4034 decimal digits. I hope that helps you.

Guest Oct 7, 2020